The tire in your photo appears to be an R1/agricultural tire. In the USA that would normally be a four-ply tire.
If money allows, I would replace tire so it does not catastrophically deflate at a critical instance.
If you must continue with it, a tire shop should be able to install a "boot" within, a tough piece of rubber/fabric cemented over the hole from the inside and bonded with heat so it stays in place. If the boot support area is decayed this is but a temporary repair. Boot is an expedient, not a cure.
I like good rubber on my vehicles, passenger vehicles, UTV and tractor. Tires are cheaper than the hospital. I replace tires "sooner" rather than "later". Equipment down time is a cost. I abhor equipment down time.
Tubeless tires are normally repaired with plugs.
After your long tractor search I know this tire problem must be disheartening.
It was really surprising to me how quickly it deflated! Though, as far as catastrophic, there wasn't much risk there, going so slow. I then drove it in reverse with the FEL supporting the front weight, using the brakes to steer, given that the field I was in was 1/4 mile from the fields behind our house, and I didn't feel good leaving the tractor that far away.
I did have a frightening experience going to Croatia on our honeymoon in a VW Transporter when a rear tire blew out. If the van had been loaded down, it would have been really bad. We usually swap tires for summer/winter on the cars, so it helps with staying on top of tire wear.
Yes, it is an ag tire. It is actually new. I could tell as well, having it in the car, from that "new tire smell", but there was almost no wear on it. e.g. the importer in Poland put it on. It has nylon reinforcement/belting. Is this pretty standard in ag tires? I think the boot should stick pretty well, I'll just have to ask around who does that.
The rear tires are Korean (Hankook), and have a only little bit of wear on them. Also 6-ply.
I am inquiring about foaming for some of the two wheel tractor and implement tires for this reason. One of the mowers has just 2 plys, and long thorns punch through it, which last year was a big problem. The equipment down time is a nuisance, even if it is a tiny 3.00-4 tire on a hay belt rake. I haven't gotten to needed repairs on the Goldoni 128 transmission either yet, which means Wednesday using the rotary plow with center of gravity shifted by the reversers, such that I'll have an aching back again, but potatoes have to be planted.
But from what I've read, foaming can affect traction, and I don't either know that I want the permanent extra weight for the Iseki.
Because of not realizing till about December that I needed the FEL, and this 4 wheel tractor, I already have the mulching mower for one of the two wheel tractors, and next week I pick up the adapter to get it to fit the Goldoni (it was a good price new-old-stock condition from Czechia, but for the Nibbi brand). So I'm not going to be mowing brush with the Iseki, just have to be careful not to go places that were brushy until recently.