Best thing that you can have on the road when changing solid type tires is an air grinder and a chisel. It the tire is ripping while being pressed off and there is no chance of getting it off in one piece then press the outside part completely off the wheel and guts of the tire. What's left is the inner part with the steel chords, usually 3-5 depending on the tire size. Just cut the chords with the grinder and for the rest cut as deep as the grinder will go. After that a chisel will go through the rubber very easily and then all you have to do is get your pry bar and separate where the tires cut and pull the rim out. You will run into this alot with your cheap tires. But even with quality tires you will run into it more with smaller ones like 5.00-8's and 18x7-8's. Using a starting cone to mount the new tire will help prevent this also as it will make sure the tire mounts without ripping any as it is being pressed. Bolt together wheels, you don't have to worry about this. I prefer lock ring type wheels over bolt togethers because you don't have the hassle of breaking down every nut and then prying the two pieces apart and then making sure you have everything line up perfect for remounting. But that's just me. For dealers and independent shops, best thing to do is leave it to the professionals. Sure you can do it, but we (tire suppliers/installers) do it everyday so it's like a habit.

Mostly we leave this to the company who sells tires, and only quote "locked in" customers who would rather be billed through us than have to open an added vendor. I would also note that a battery powered saws-all is a wonderful tool to have for those hard to remove "solid pneumatic" tires, just cut them to pieces.
And no matter what you do, there are always going to be times you have to come back to the customer and explain the added difficulties and related expense that results from some choice the customer made previously, like having purchased a poor or the cheapest tire for their operation and not having stopped the truck when changing the tire would have been far easier. I think you just have to be as diplomatic as is possible, and have the salesman who has a relationship with the customer's purchasing department deliver the "bad news", rather than putting it on the service department's back.