Well, me giving advice on this is about the same as the blind leading the blind, as I've done a scant few of these in my life, but here goes.
Once deflated, the pain in the arse part seems to be breaking the bead from the rim. The next thing that will be a pain (and you must have pair of them) is using the tire spoons to work the bead over the edge of the rim. I've watched tire repairmen do it and it seems effortless, for the rest of us who don't do this everyday, not as easy. You want to use those spoons to your advantage, take small sections at a time and work them gently over the rim, you can nick or tear that leading edge of the tire if you take too much, just feel your way through it, if it don't feel right, it ain't as I see it, take less of a bite and work what you can, it's all in the way you use those tools, seems easy for experienced hands. Another must is some lube, you want to make that bead slippery, something like rim glyde from Napa or similar, some use soapy water, I like the rim glyde from napa, does the job and does not make a mess. I stand to be corrected here, but last fall I tackled a CACL loaded 13.6 x 28 tire with no problem at all, I used the bucket on the same tractor that was stablized with counterweight on the back and blocked up safely, a short piece of 4x4 lumber to press down on the tire and break the bead, I put a sheet of plywood under it to keep it out of the crushed stone driveway. TSC has the spoons, I got one regular and one for a drop rim, latter has a bend like kind of like an S in it, not knowing which one I would have liked a pair of having not done this before, next time I will get one more of those for a drop rim or whatever you call it, the one with the bend in the working end, seems to work nice on regular rims, shape seems to be just right, and does not seem it can pinch a tube if you use care.
Thinking back, the methods in which you use the pair of spoons is really the key to the whole thing, breaking the bead is easy enough if you have the means, loader, tool, or something without having to pound on it. Placement of the spoons where you pry from, on each other, sometimes I used a 3 lb hammer to hit em on the side to work em in, same action as a tire machine sliding around the rim, I'll go out on a limb and say knowing how to use a pair of those spoons is really the secret to it all, it was the most time consuming part of the job for me as I recall. Gemplers has a pictorial view of changing a tractor tire on their site, check that out, it's on a 2 cyl. JD tractor.
Take a look at Gemplers web site, they will also have everything you need, they also have a material that lubes and keeps the tire from bonding to the rim, all sorts of tire spoons, even a bead breaking tool, may cost you, but you'll most likely need something to break that bead that you don't have, hard to pound one off sometimes, you need that down pressure and weight as I see it, the loader was real handy for this and did the job quickly, just have to carefully apply the down pressure, don't over do it, it squashes down and 1x-2x of this did the trick on each side. When you re-mount the tire, you need to use those spoons again though the first bead is easy, then you put in the tube, Gemplers has spoons that won't pinch the tube, just use care when doing that, once you work the 2nd bead over, inflate, then deflate to work out any wrinkles in the tube, that is about as best I can describe my very limited experience with it, I asked here just like you, then did it, but was proud that I completed the task myself, it did save me some money, and I realized that it's easier done then said.
Other than something along those lines and if you don't want to get involved with the above, take it to an ag tire shop, will at least save the road service call, and you won't have to buy the tools or source a loader or similar to break the bead, (though you could use another wheeled tractor or maybe a truck to do it by running over the edge, not sure if that can damage anything in the tire) tire shop most likely won't damage the tire, and it will be done by experienced hands.
