General Assumptions:
Just because ~255 ships were "transferred to SLN control" does not mean ~255 ships were put into service in the next 4 years. With this as my basic assumption, I've not constrained myself to 252, and based on TRO 3057 Revised, I struggled to keep it under 400. I did settle upon 385 (including transports). I justify this with comments, below, but also on the fact that the writers of H:RW clearly didn't do their math or check their sources when they had the AFFS field more than 3 Davion I destroyers in 2577. And that's just one example of where they went wrong, and not nearly as entertaining as some of the errors in Handbook: House Liao.
Ships:
Vincent: 36
Sure, 530 would eventually be built, and it'd been in production for 100 years, but as the ships are worthless for anything but picket duty or convoy escort, the HAF has no need for large numbers; plus there will be 39 versions in 300 years of production, so despite H:RW calling them Mk 39s, RW-era Vincents are probably more like Mk 10s or 15s. Worse for the numbers of the Vincent, the HAF is specifically said to use Frigates for picket duty, rather than corvettes (probably due to the poor performance of the early Corvette designs). I could see actually see the HAF passing on this design entirely, with all Vincents in the original SLN fleet coming from the Great Houses, except that the Vincent profile does claim they replaced the Nightwing, Tracker, and et cetera crappy (or rather, crappier) corvettes. There's plenty of time to build more when the SLN is responsible for internal security of the entire Inner Sphere later.
However, these unsophisticated ships would be prime contenders for something the HAF would be likely to export to other nations, so the SLN may have ended up with castoffs from other nations to boost the actual percentage of Vincents in the SLN.
Essex: 24
There's so little known about this class that I feel justified in limiting its numbers.
Baron: 48
"Over half" of the (surviving) examples of this class will be donated to the Great Houses in 2582, alongside the rebuilt Aegis cruisers. But this didn't stop H:RW from deploying both Baron and Aegis class ships in the AFFS in 2577 for Case Amber...oops.
Lola I: 16
It's a heavy destroyer, seeming more along the lines of a flotilla leader than anything else, so it wouldn't need production levels on line with other destroyers. I'd actually call it a light cruiser, myself. But either way, it's clearly not replacing another ship class, just augmenting the fleet as a whole.
Avatar: 80
If there were 106 Aegis cruisers to retire, there were probably more like 120 built. I'd rather build 120 of these for 1-to-1 replacement, but 2-for-3 will have to do because of the silliness of H:RW
Riga: 60
18 months to build, in production continuously for 102 years per TRO 3057, that gives us 68. Assume 8 were lost in the Age of War.
Congress: 20
It's just entering production, and if it takes as long to build a Congress as it did a Riga (18 months per), only 20-25 would be in service with the SLN at the time of the RW.
Quixote: 25
250 built, and 3057R says they were replaced by the Congress rather than the Riga. 10% still in service seems minimum, given the numbers of the Riga and how few Congress are in service; 60% is likely more realistic, but see comment to Avatar-class.
Monsoon: 48
"Only" ten were modified with LF batteries later, which suggests that it was a fraction of what the HAF/SLN had. Let's call that fraction 25%, and assume some Monsoons were lost in the RW and Hidden Wars.
Farragut: 8
Not only are they said to replace the Dreadnought class (of 4 ships), but the Texas which followed them had a similarly small production rate. There were 52 Texas-class and 280 McKennas, so I'm assuming a similar proportion between Farragut/Monsoon.
Carrack: 20
These are specifically mentioned as having been a HAF design, so it stands to reason some should be around. The Star League probably manufactured more, but 20-ish would do a good job of moving military cargo without the need to rely upon civilian shipping.
This would break the structure of the Hegemony Navy down as follows:
Corvettes: 36
Destroyers: 88
Cruisers: 80
Frigates: 110
Battleships: 56
Transports: 20
Plus whatever they got from the Great Houses which was actually useable, which may or may not include the Davion Is, depending on how one interprets the profiles of the Whirlwind and Davion I.
Re: Battlecruiser designs, I do notice that the Quixote-class were supposed to be Battlecruisers.