As far as laying it on its' side, you can probably do that if you remove the air filter and the battery. Remove the battery so it doesn't leak acid and remove the air filter so it won't get soaked with fuel or oil.
It probably will leak oil or gas though.
If it was me, I would just remove the transaxle, fix it and reinstall it. On most mowers its usually just a belt, 2 control links, and about 6 bolts that hold it on.