Re: Is this an "A" type OD?
Actually, the 4-synchro gearbox itself is a bit longer, to accomodate the extra synchro ring. It's not just the flange thickness that causes the rear mounts to move.
Definitely an A-type, definitely a 4-synchro box. Should work fine in a TR3, but as Don notes, you'll need to slot the holes in the rear crossmember (I had to go more like 3/8" to eliminate all force on the mounts). With a 56 TR3 (or TR3A before TS50000), you will also have to massage the transmission tunnel slightly, to clear the starter bulge. Possibly you'll also have to install longer studs in the rear of the engine block, but I don't think so. I did for the late TR6 box I used, but it's even thicker than the TR4.
FWIW, it is possible to rebuild the TR3 top cover with the TR4 shift rails & forks, to retain the dipstick. Or the TR4 top cover will work fine too, you'll just have to use the plug on the side to fill the gearbox or check the level.