mtncrawler, good photo of the wood placement!
Have you guys considered using "SLIME" or some commercial tire sealers? You pump these products into the tire by removing the valve core, releasing the air, then pumping an appropriate amount of sealer into the tire, re-inserting the valve core, and then re-filling with air.
SLIME is probably the best known brand. Ag suppliers like
www.gemplers.com sell a couple other products. UltraSeal Commercial Grade and UltraSeal Bulletproof Grade. The Bulletproof grade is US Military approved and will seal a hole up to 1/2" in diameter while the Commercial grade is good for 1/4" diameter hole. Gemplers sells it by the gallon, and I'd guess one gallon would treat 8 to 12 tires on a snowcat, depending on size.
All of these eliminate slow leaks and most puncture problems.
Snow Tracs & Snow Masters have a somewhat different problem because they use a series of small wheel held on dual (or triple) leaf springs. Villi, one of our members from Iceland
showed us his "Villi Tool" that is used for changing tires the small bogie wheels on a Snow Trac/Snow Master. It could be adapted to other brands that have small bogie wheels. I also made a couple of these tools out of scrap steel. I believe "Mith" made one for "Ice Queen" too. Here is a link:
http://www.forumsforums.com/3_9/showthread.php?t=2460
Below are some photos of his tool: