What Small Craft designs are used for Marine Boarding Ops?
First off, my first advice is to accept that you're not going to find anything near as fast as the Battle Taxi. It's just not gonna happen, sorry. In addition, a lot of your other options are going to have less firepower. Fortunately, guns are 100% irrelevant to a boarding shuttle, as there's no way to actually carry enough to make a difference while still carrying a decent amount of troops.
Here's some options:
Debuting in 2438(XTRO Primitives II) or just after are the first three Marks of the Ares. They're slow at 3/5, but getting them to the target is pretty much the DropShip's or WarShip's job.
Mark I is the serious hauler, 65 tons of marines should be enough to capture just about anything, guns are comparable to the Battle Taxi, and while it has less armor(281 points), it's still enough to survive a few passes by enemy heavy fighters.
Mark II is the gunship, with enough firepower to make fighters and even small DropShips flinch. Can still carry 15 tons of troops.
Mark III is a mix between the two that also boosts the armor to 326 points. Only carries 2 platoons of troops, though. My least favorite of the bunch.
After this, we go back to TRO 3057r. There's three that I'll point out here, all move at 4/6, can carry at least a few platoons of troops, and have enough armor to survive a few incoming salvos.
In 2528 we get the ST-46 Shuttle. Completely unarmed(don't care), 320 points of armor, and 12 tons of space.
2536 gives us the K-1 DropShuttle. Just enough guns to make the lightest of fighters a bit nervous or be useful as a customs/police boat, 360 points of armor, 16.5-ton bay.
The Clans would debut their upgrade in 2914. The K-1C has pretty much the same cargo, better guns, and boosts the armor to 444 points.
This is 100% a tooting of my own horn, but the K-1 series has what I feel is a rather nice *otW article here:
https://bg.battletech.com/forums/index.php/topic,73453.0.html :)
Finally, there's the latest iteration of the 'Mark' haulers, the Mark VII. (The IX is still more recent, but you're asking about stuff that predates the Taxi.) Despite being the only military-spec vessel of this trio, it has the thinnest armor at 'only' 232 points, which should still be enough if your DropShip pilot is at all competent. Firepower is slightly better than the K-1 and more focused due to the aerodyne shape. 31 tons is a LOT of troops, second only to Mark I. A 2841 release date means this ship comes too late for Star League usage, but would be the New Hotness during the fierce space battles of the early Succession Wars.
Like the K-1C, the Clans' Mark VII-C improves the guns with pulse lasers, and the armor is now 264 points. Same intro year, which tells me this was either a case of parallel evolution, or the Mark VII was already on the drawing board when Kerensky left and the specs went with him. Or it's a mistake. We do have to remember that TRO writers are human, after all.