Most of the hydro drive lawnmowers I've seen have sealed 'non serviceable' drives. If you can't find documentation or other advice, and resort to guessing..( if it were me.. ) I'd use the hytran type UTF. It is pretty much a 20wt oil with lots of addatives anyway. The ATF may be thinner.. but not much.. it will however be high detergent.
The UTF covers such a wide range of uses.. power steering, hydro trans in tractors.. wet brakes, hyd's, manual tranny, final drives / gear reduction units.. etc.. ), that it is at least a decent gamble.
If you find any specific info.. post back. Who makes that machine? MTD toro? etc...
Soundguy