Why did the Star Kingdom not aggressively seek attritional engagements to wear away the Peeps' mobile strength?
What makes you think the Star Kingdom didn't?
In fact, that was precisely what the Star Kingdom was doing -- rather successfully, too -- before the operational hiatus set in after the successful conquest of Trevor's Star. By the time Trevor's Star had fallen, the Star Kingdom calculated that it was getting very close to attaining -- if, in fact, it had not already attained -- numerical parity with the Peep main battle fleet, which clearly would have equated to strategic superiority, given the Manties' advantages in weaponry, tactical doctrine, and the general capabilities of its officer corps. There were still those damned battleships to worry about, but the general perception was that the Star Kingdom and the Manticoran Alliance were controlling the tempo of offensive operations, and that the ratio of losses was persistently in Manticore's favor. This was the classic realization of the prewar strategic doctrine. For reasons I've already discussed above, attacks in force on first-tier and second-tier star systems of the People's Republic were not viewed as a viable strategy. (And, by the way, I suppose I should add that the reason the Manticorans didn't view it as a viable strategy is that, given the constraints I'd set up, I didn't regard it as a viable strategy.) Trevor's Star and Barnett were exceptions to the rule, in some ways, but, by and large, the Alliance had been successfully pursuing its prewar strategy and gradually winning the war.
The fact that you didn't see all of the battles involved doesn't mean that they weren't taking place. Most of the battles you did see were engagements in which the Peeps' losses were both proportionately and absolutely worse than those suffered by the Manties and their allies. There were other battles which followed the same general pattern.
In the attack on Nightingale, where Hamish breaks off the action when the Peeps spring their ambush on him, there was no need for him to continue the attack. The strategic object of capturing the Nightingale System wasn't individually that important, and the significant point about the battle wasn't that he would have inflicted heavier losses on the Peeps than he would have taken (which was undoubtedly true, despite the tactical situation he faced), but that the same losses could be -- and were being -- inflicted on the Peeps for a lower cost in other engagements where the tactical situation wasn't as unfavorable as the one in which he found himself. In other words, he wasn't afraid of taking losses in return for inflicting losses; he was unwilling to take more losses than he had to to inflict losses on the Peeps.
The hiatus that sets in after the conquest of Trevor's Star is the result of the fact that the Manticorans had run their maintenance cycles so far into the red that they were beginning to suffer significant system reliability problems. In addition, as has been pointed out both here on the Bar and, I believe, in the books, the introduction of new technology had been taking place on an "as available" basis. New construction was commissioned with the new systems, but older construction received them only on a sort of patchwork basis as ships were returned to the yards for maintenance or repair. And given the tempo of operations at the front, wallers weren't being returned to the yards except for major repairs which couldn't be taken care of in forward areas. This meant that the capabilities of the individual battle squadrons were not homogenous, which had a significant impact on the tactical flexibility of the RMN and its allies. So, given the fact that it was absolutely critical to recall ships in the worst need of maintenance for refits, and that those refits would also offer the opportunity to refit with new compensators and other technology advances (including the FTL com), the Manticoran Alliance chose to revert from an offensive stance to a temporary defensive stance, with the unstated strategic understanding that standing on the defense would allow them to continue to inflict disproportionate attritional losses on the Peeps if the Peeps decided to press the offensive.
Unfortunately for Manticore, what the Alliance got was Esther McQueen. The Manties' willingness to revert temporarily to the defense was based on the strategic performance of the People's Republic up to that point, which had been significantly handicapped by the interference and heavy-handed "security tactics" of Rob Pierre and, especially, State Security. Virtually everyone on the Manticoran side -- including Honor Harrington -- were unprepared for the sheer audacity of McQueen's Operation Icarus. Icarus was the first major Peep offensive operation in which the Alliance's losses, just in terms of mobile firepower, were proportionately worse than those suffered by the People's Navy. The economic damage inflicted at Basilisk tipped the balance of achievements for Operation Icarus even more heavily in Haven's direction, and the secondary operations and the effect they had on Allied morale were, in many ways, even more significant. In essence, the temporary defensive stance which the Admiralty had been prepared to embrace got transformed into a much longer period of defensiveness because of the political necessity of reassuring the Star Kingdom's allies -- and the Star Kingdom's own public -- that the Peeps would be unable to repeat Icarus anytime soon.
Ultimately, that delay worked significantly in Manticore's favor. First, because it gave additional time for the new weapons systems -- especially the MDMs -- to be deployed in decisive numbers. Second, because it gave time for the tension between Saint-Just and McQueen to spill over into McQueen's unsuccessful (and defensive) coup attempt. This may have brought Theisman in to command Capital Fleet, but in the meantime it meant that when Hamish launched Buttercup, the command structure of the People's Navy had lost its best and most charismatic strategist and was in significant disarray, which made the advantages provided by the MDM even more decisive. And, of course, in the end, the fact that Theisman had been recalled to Nouveau Paris eventually brought about the complete fall of the committee of public safety and the restoration of the Old Republic. Had it not been for High Ridge, on the Manticoran side, and Giancola, on the Havenite side, that would have decisively ended the war between Haven and Manticore. As it was, what it resulted in instead was the resumption of hostilities between the Star Kingdom and a vastly more powerful and internally coherent Republic of Haven. Talk about your law of unintended consequences.