Elizabeth's new royal yacht, HMS Duke of Cromarty
Yes, that's right: Elizabeth's new boat is based on a battlecruiser design. But you're wrong about their converting an existing ship. They thought about that, but the difficulties in refitting warships apply to transforming them into yachts, as well, you know. Also, the Cromarty was ordered longer ago than you seem to be assuming; it's just not going to show up until the chapter we've been discussing in whatever I decide to call the book. (IRTW, I've been hardening on At All Costs, although I'm not absolutely wedded to that title just yet.)
In fact, the new ship was authorized and ordered in the very early days of the High Ridge Government. Obviously, a replacement was necessary, and ordering one quickly -- and making the point that it would be capable of looking after itself in a pinch -- was one way for High Ridge and his cronies to make themselves look good and capable of decisive action in the early days. And, in one of his relatively few really bright moments, it was (shudder) Janacek who came up with the obvious solution. Said solution was to take one of the Agamemnons which had already been laid down and simply convert a portion of the missile core into additional personnel space. That's a lot of volume, and there was plenty of room to provide space for a couple of hundred passengers in really comfortable staterooms, conference rooms, even a formal ballroom, without a great deal of difficulty. Basically, they used about half the potentially available volume, so Cromarty is essentially a fully armed and capable Agamemnon with its ammo allotment cut in half. It carries the full crew of a standard Agamemnon; it has all of the same defenses (and Keyhole); all of the same energy and chase armament; all of the first line sensors, ECM, and EW; and enough missile pods to look after itself while running away from any potential threat. As yachts go, operating this one is going to be pretty damned expensive. On the other hand, you can probably save a little bit on escorts, and it's about as tough a ship -- and as safe and secure a means of royal transport -- as you're going to get short of a superdreadnought.