29Y01311 serial number tells me the tractor is early 1983 year of build, and it's fitted with the 3F/3R transmission.
The earlier 931B's only had a 3F/1R transmission, but the 3F/3R came in as an option around the late 1970's.
Virtually all 931B's had dry steering clutches up until about 1986, but a conversion to wet steering clutches could be purchased from the early 1980's.
I'm not sure how many 931B's would have been converted to wet clutches, but I'd hazard a guess that it wasn't many.
If the steering clutches have been converted to oil, they will have a fill point and oil level dipstick.
There is no way that transmission oil could get into the bevel gear compartment. It is a sealed compartment (apart from the breather on top), and no transmission lines run into it, or through it.
However, the transmission oil is used to actuate the steering clutches via hydraulic pistons inside the steering clutch compartments.
The oil lines run from the top rear of the transmission to the valves behind the foot pedals, and thence to the steering clutch pistons.
The transmission should not use the same viscosity oil as the bevel gear compartment. The transmission should be filled with a Cat transmission oil (TDTO or TO-4) rated at 30 weight, and the bevel gear compartment should be filled with TDTO or TO-4, 50 wt oil.
You can also use a regular 80-90 wt gear oil in the bevel gear compartment, as used in most automotive rear axles.
I'd like to see a photo of the underside of the rear main housing (where the bevel gear and steeering clutches are located), and the plug identified, that you are draining the "excess" oil from.
If the transmission is only working for a few minutes, then it is running out of oil pressure, and is likely running out of oil, which causing the loss of oil pressure. The transmission needs oil pressure to keep the clutch packs engaged.