Overly simple, a new “regular” Shimano 12-speed HG-type cassette (not the microspline) should fit on the existing/stock freehub. But, as Shane found in his H4 review, the freehub maker they chose made them in an incompatible way with bad tolerances. Some handy work with a file might get you there with the stock H3 FH, but it sounds like it’s not plug & play like it should be.
Yes, Saris has offered and XD/XDR freehub for many years. Despite that, I have seen them mention on more than one occasion that even with that FH, the trainer is not 12-s compatible. Presumably that would imply they designed it to work with the XD/XDR that were 11 speed, in that time before Eagle on MTB in particular (and whatever on the road side that I didn’t follow as much, but maybe the Apex 1x?).
To the general 12 speed compatibility on the trainer, people have reported decent function until you get to the largest cog (lowest gear) as you get the derailleur close to the trainer frame. If I heard right, they altered the axle mounts on the H4 and shifted the mount points over 3mm or so, to provide clearance. Presumably that might be just new end caps, but I am guessing at that.