Personally I would not.
The Pro 44 has Hydro Gear ZT-3100 transaxle with a maximum gross weight limit of 1200 lb [544 kg] so I think you, the mower (789 lbs+), and smoker will definitely be more than those transaxles can handle safely.
And Gravely has those transaxles listed at $1,336.45 - $1,341.95 a piece depending you have to replace the LH or RH units if not both.