We had that problem too..
We had a flat every day on the torch car at the junkyard too..always ran over bolts,steel scraps,etc..but we had lots of spares!..so we'd throw another tire and rim on..solid tires are nice if you can get them,they are costly new though,and the yards I've seen use them say they suck for traction..only work on heavy vehicles best..we tried everything to get around having flats..the best way was to watch where your driving,and pick up the shrapnel that punctures tires often!..
We had a few army trucks with "non directionals" on them..those things were like iron,had 10+ plys..I cant recall them going flat from a puncture,but they were dry rotted and porus,and leaked all the air out in a few days..tire sealant helped slow the leaks down some...those would be good if you can get some..only come in 16" though I think..
We tried making "paddle wheels" out of rims by welding angle iron on them once..they were not the hot setup..too low to the ground with no tires,an all they did was dig holes!..we tried filling tires with cement once,but they weighed almost 200 lbs!..and the cement broke into nuggets and turned to dust quickly..
Hardware store foam wont be strong enough to hold the weight up..I've tried that on riding mowers,and it wont even work on them.

.that foam they use in loader tires is special stuff..
A lot of yards tried everything to solve the flat tire problems..one place tried putting a smaller tire inside a larger one to prevent puntures..it did help some,,and it was a bitch to mount the "outer" tire!--they used a tire with an inside diameter about the same as the outside of the one on the rim..stuffed the smaller one inside,mounted the rim,and inflating it wedged it against the outer tires inner surface..once water got between them,they often spun,and the vehicle was stuck..

back to the drawing board!..
Another brainstorm was to use tubes,and fill the tires with mostly water ,rather than all air..they seemed more puncture RESISTANT,but not bullet proof...
..if you could get enough RTV silicone sealer/gasket maker,I bet that would work,but it'd cost a lot of money..might as well have the tires foamed!..
We used a truck for our torch car towards the end,it was an International 4x4 with 16" tires..we found 4 tires rated "10 ply" that were bias ply,on an old ambulance..they were very stiff sidewalls,and I think only one of them got puntured..and we ran over LOTS of things that should have punctured them..
One yard said they had the best results on one car they used ,by putting several sets of tire chains together,so they had a "cross link" on every link!--sort of "flexible armor"..gave nice traction too..but they only could use them in the rear,they caused a lot of problems up front..another expensive,and PITA to fix option..there is no real easy solution that I know of,other than the foam..
We were much more concerned about the LOADER having a puncture!!..then your talking about 800+ bucks!!

--and dead it the water without it!..we used huge tire plugs with a patch made onto them made by TECH..they were 1-1/4" in diameter,we had to hole saw out the punctured area and install the patch..and pray it held!..we installed one in less than an hour one snowy day in 2003..
Maybe if you put 100,0000,000,000,0000 BB's in each tire..or lots of golf balls...
