I am not familiar with this machine but I gather it is self propelled?
If the above is true and you just need to restore some lost traction you can resort to equipping the slicked out tires with chains.
Now this is just my suggestion for how I might deal with the dilemma if I was in your shoes and needed some traction.
If the wheels that the tires are mated to are of "disc wheel" construction (like standard auto wheels) and not "spoked wheels"...............I would do the following.
Obtain some link chain having 1/4" dia link thickness. How much chain length you need will be determined by the length of the individual chain segments needed to outfit the tires.
Pick a point on the wheel disc that is about 2" from where the disc section of the wheel meets the tire rim section of the wheel. Drill a 1/4" hole through the disc at that point.
Now, using a 1/4" bolt (you will have to decide how long it needs to be) fitted with a flat washer big enough that it can't fall through the chain link.......put the bolt through the end link on one side of the wheel and stretch the chain over the tire tread until it meets up to the drilled hole and bolt on the opposite side of wheel.
Make sure you can get the bolt through a chain link. Enough bolt must come through to get a flat washer and a locking nut onto the threads. Mark that link and cut the chain at the next link BEYOND that marked link. You can now use this segment of chain to measure the rest of the chain segments you will need.
You will need to fit a chain at 8 points of the clock face in order to have a good working traction system. 12 o'clock, 1:30, 3 o'clock, 4:30, 6 o'clock and so on.
The drilled holes at 1/4" diameter will not cause problems.
Be sure to use nuts that are self locking or alternatively use a good strong thread locking compound on the threads.
Or if you have enough excess thread showing you can put an additional nut next to the first nut and jam wrench them together for a non-loosening configuration.
If the wheel is a "2 piece" set up "having inboard and outboard halves) held together with bolts, you may be able to use some of the OEM bolt holes instead of drilling all new holes.
Having a photo of a wheel would help.

Interesting ideas! The wheels are spoke, but I'll bet I could anchor the chain around one spoke and wind it around the tire around the entire circumference of the wheel.

Thanks for the idea.

This is what i did to a Troy-Bilt walk behind mower with the large solid rubber rear tires that wore slick. I removed each wheel, took an old nobby bicycle tire, cut the bead from both sides, cut the tire to fit over the wheel. Put construction adhesive all around the wheel, anchored the bicycle tire to the solid rubber tire with 1 inch galvanized screws. They are still usable after three years. My neighbors call it a "Redneck Recap. Works for me.

The redneck recap is interesting, and very "do-able" but, I ran a 1 ton, 2 wheel drive "dually" truck in some pretty un-a-TRACTIVE terrain when I was selling firewood back in the early '80s. Had a set of chains I made that covered both tires of each pair.
Went places that friends had to use their 4wd to negotiate, so I am still a firm believer in chains for traction.
Tombob, I believe you can do exactly what you said and you will be just fine.