This is a really interesting thread; I'd be a little suspicious that the material in the mouthpiece itself is responsible, but it is possible I guess. I like the idea that the rental companies are using a disinfectant that could be the cause of the irritation, it just makes sense to me. Certainly, I could be dead wrong and maybe you do have an allergy to the material that typical regulator mouthpieces are made of. Maybe you can find out exactly what that material is...I bet it's not pure rubber or latex. Would Scott Zeagle have an answer for that? Until you find this answer, it doesn't make sense to me to invest in a regulator set, as you may have the same problem.
Regarding the sea cure, it is a very comfortable mouthpiece, but it's not 100% silicone. It's actually a fairly stiff plastic. You'll want to find out what it's made of; probably the same sort of material that athletic mouthguards are made of. A dentist might have some info about this, especially one who deals with athletes.
Then, there is the question of why your lips get irritated and not your gums or any other part of your mouth that the mouthpiece is in contact with. This is correct, right? If so, it seems to support the idea that dehydration may play a role in it. One difference between breathing off a reg and a snorkle is that the air from a reg is MUCH dryer. This could be a factor as much as the mouthpiece material, I suppose.
Anyhow, when you get it figured out, then you can consider a reg purchase. There are really good ones for around $200; zeagle envoy, SP MK2/R190, aqualung entry level (whatever it is) any of these would work fine for recreational diving. There's also nothing wrong with buying a used reg, and some of the older metal case 2nd stages are terrific for relieving drymouth. The only thing is you need to bring the reg to get it overhauled before you dive with it, and this adds to the cost, plus it's important to find a good service tech.
Good luck, and please post on here when/if you find some answers.