Why doesn't Elizabeth pack the Lords?
Faith is the heir to Honor's duchy. As such, she would be the person who would sit in Honor's cadet seat. Unfortunately, she is very much a minor, which means she isn't legally competent for a seat in the Lords at this time, so the Harrington Duchy cadet seat is empty. Devon Harrington has an heir, who is currently sitting in his cadet seat. That fact hasn't been significant to any of the novels, so it hasn't been mentioned.
In answer to your question about whether or not Elizabeth could effectively "pack" the Lords by appointing nothing but earls or above for the San Martino peerage, thus effectively giving each of them two votes because of the cadet seat system, the short answer is "no." The long answer is "hell, no."
I'm going to go into a little more detail than you probably expected, but that's partly because I'd like to kill several avians with a single flung chunk of mineral.
The terms of the Act of Annexation are specifically drawn to require that the San Martino peerage be "comparable in all respects" to the peerage of the pre-annexation Star Kingdom. Within the context of the Act, this is clearly understood by all parties to mean, among other things, that the same general proportions of the various titles will be maintained in the new peerage. Moreover, the House of Lords would (quite rightly) unanimously vote to exclude all of the new peers if any such attempt was made, and despite the Crown's control of the judiciary, the courts would no doubt support them in that action. And finally, if she attempted it, and if the House of Commons was prepared to go along with her, public opinion -- and the electorate -- in the Star Kingdom would undoubtedly recognize that in doing so, the Lords were acting to defend the prerogatives guaranteed to them by the Constitution against a deliberate encroachment. That's the short reason why it isn't going to happen.
The longer reason comes in two parts. First, the House of Winton has never been in favor of packing the Lords, and Elizabeth as an individual isn't either. She is completely willing to select new peers she believes will support the policies that she wants to see enacted, but she is not prepared to do so in a way which establishes a clear precedent by which the Constitution can be subverted by future monarchs. She is a constitutional head of state, and she has no taste or desire for becoming a dictator. The House of Winton has pursued a gradualist approach for centuries, and Elizabeth, despite her frequent frustration with peers like High Ridge, has a powerful sense of history and respect for precedent-based stability.
Even if Elizabeth was willing to undertake such a maneuver, the Commons would hardly be likely to go along with her.
First, the entire domestic political structure of the Star Kingdom is built around the concept of the primacy -- though not the supremacy -- of the House of Lords. It's the way that the political leaders of the Star Kingdom have been habituated to think.
Second, the situation as it stands at the moment is one which heavily favors the Liberals and Progressives, and it is entirely due to the majority they've been able to assemble, along with the Conservatives, in the House of Lords. A key point which I think a lot of readers are missing is that the positions that the High Ridge government has taken are genuinely popular domestically. Until the announcement of the existence of Thomas Theismann's new fleet, there was absolutely no reason for the Manticoran man in the street to think that the Star Kingdom was in any danger, realistically speaking, from the soundly defeated, technologically outclassed, civil war-ridden, Peep Navy. There was nothing that Navy could possibly do if it came back to a shooting war between the Star Kingdom and the People's Republic... excepts to get itself blown out of space. The voters might not be happy about the ongoing tax burden they faced, but most of them were willing to continue to bear it so long as they believed that at least a significant portion of the total money collected was going towards maintaining the military supremacy upon which their sense of security depended. Moreover, even many who might have objected to the use of that money on purely military spending were willing to continue to put up with the taxes because of their understanding that what was really happening was that the money was being spent on domestic programs of which the Liberal and Progressive Parties strongly approved.
Third, it is significant to remember that for the past half-century, or so, the Centrists have been gaining ground steadily in the Commons, to the point that all by themselves, they represent the majority party in the Commons even at the opening of War of Honor. During that time period, the mainstay of the Liberal and Progressive (not to mention Conservative) influence on government policy has been those parties' representation in the Lords, not in the Commons. In fact, the Commons has been seen as the chamber of Parliament which was the natural enemy of those parties when they formed the Opposition. Even in the worst periods, they were still able to form a very strong minority voting bloc in the Lords, but that was not the case in the Commons. What this means is that there is for all intents and purposes a generation of Liberal and Progressive MPs who have come to regard the power of their parties in the Lords as the true check on the policies of the Centrists, Crown Loyalists, and the Crown itself. The theorists and ideologues in the Liberals would certainly support (in principle) the notion that the Lords should be abolished, or at least considerably reduced in power, on the basis that the Commons represents the best expression of the will of people. The Progressives have no problem with the notion of a powerful hereditary aristocracy to begin with. Their social policies have always been much more "pragmatic" and far less concerned with the notion of the redistribution of power or wealth which have been so central to the Liberals' ideology for so many decades. So what we have here is a situation in which the Conservatives have no interest in all in reducing the power of the House which is their natural haven; the Progressives have no interest in shaking up the social order, because their political agenda is much more directed at maintaining a sort of "fat and happy" proletariat which does as its told while they manage the foreign economic and military policy of the Star Kingdom in a way which will maximize the Star Kingdom's wealth and minimize the threat of entanglements or war with foreign powers; and the Liberals, in many respects, see the fact that they were able to survive the Centrists rise to ascendancy at all as an indication that the "checks and balances" the Founders designed into the Constitution worked exactly as intended. The existence of the Lords, which preserved a solid cadre of Liberal leadership which didn't have to worry about being reelected, allowed the party (and its ideology) to remain a vital force (as they see it) in the political agenda and domestic policies of the Star Kingdom despite the fact that they were steadily losing ground in the Commons.
The Centrists and Crown Loyalists, on the other hand, have had exactly the opposite experience. Obviously, the Crown Loyalists, whose essential article of faith is that a strong monarchy makes for a strong Star Kingdom, would see the continued resistance of certain groups in the Lords to the policies of the Queen and her Chief Minister as a reason to curb the power of the upper house. The Centrists, on the other hand, have spent 60 or 70 years building up a strongly ascendant position in the Commons, even as the head of their party and the Queen's Prime Minister has been forced to rely on a frequently fragile, often razor-thin majority in the Lords, which in the final analysis was held together more by the personal authority and alliances of the Duke of Crommarty than by the undeniable popular support which their policies enjoyed. As a result, these two parties -- which are now the ones out of power in the Lords entirely -- are the ones whose members have a genuine interest in transferring the power of the purse from the House of Lords to the House of Commons.
In short, in many ways because of the political experience of the Star Kingdom over the last century or so, the parties which one might expect to have the greatest interest in breaking the power of the House of Lords, have, instead, come to see themselves as having a very strong vested interest in maintaining it. The fact that the Liberals, in particular, are steadily gaining in representation in the lower house based on might eventually come to change that perception, but it hasn't happened as of the beginning of this novel. That doesn't mean that there aren't individual Liberals who feel strongly that the power of the Lords needs to be reduced. Among the Liberals who feel that way, Cathy Montaigne is probably the prime example, despite the fact that she herself was once seated in the Lords. Or, perhaps, because she was once seated there.