You haven't told us, but I'm assuming you have ag, R1, tires on your tractor instead of Industrials. If so, I'd bet 3 or 4 gallons of
Sierra brand antifreeze would be plenty in your area. Sierra is propylene glycol instead of ethylene glycol and is not toxic. It normally sells for around $80 for a 6-gallon pack at many hardware or auto supply stores.
To fill your tires, rotate your tire so that the valve stem is at the top of the rim, and then remove the valve core to let out the air. Use a
tire fill adapter for easy filling with water from a hose. To get the anti-freeze into your tire, cut a 3' end off an old garden hose and stick a funnel into the cut end. Tape it to make it secure. Attach the garden hose to the fill adapter and pour in the anti-freeze. As the tire fills, you'll need to relieve pressure with the adapter bleed outlet so you don't get a lot of bubbling back into the funnel. Do this before adding any water to the tire.
When you have all your anti-freeze in the tire, remove the funnel adapter and hook up your regular garden hose. Start filling with water. Stop filling once in awhile and press the relief valve on the adapter to allow air to escape from the tire. When you get to the point that you press the relief and only water comes out, you are done filling. Replace the valve core and fill the tire to the desired air pressure using a compressor as normal.
Now, your water inside the tire will be over the top of the rim and it will not rust much if at all. You don't have to worry about this at all for probably the next 30-40 years. You do not need to add a tube to your tire.
Repeat for the other rear tire.
Edit: Oh yes, you may still have traction problems, especially when you are pointed downhill and try to back up with a full bucket. I'd get a heavy boxblade or other heavy implement to carry on the 3PH. Unless 2wd tractors are really big ag tractors, they will always get very light in the rear with a loader. The more rear ballast the better when doing loader work. Carrying round bales on a bale spear can get to be a lot of fun on a tractor like yours. Always try to NOT go downhill when carrying a heavy load on the FEL. Backing down the hill is much safer, but even then, you have no brakes if the rear wheels become airborne.