Use some universal tractor fluid. Kubota UDT is one but something similar is fine.
If it is really cold where you’re at then you may want to get something in the synthetic world but you don’t have anything in your car’s gear case that is synthetic and it probably is fine?
The hydraulic pump pulls from the same sump is the only concern that I have...but I live in Central Texas so below 40F is too cold for me!