For $4500 you could have a world class sounding vehicle. I am not sure what went wrong with your tweeters, could be they set the High Pass Filter on your tweeters wrong, or not at all, maybe the Bit 10 freaked out and sent a full range signal to them, I'm not sure but that really sucks that this happened to you.
Personally I don't know why they went with the AC unit, as for not much more money you could get a proper processor (like the bit 10, but not). I really hate when shops do this to people, not that they did it on purpose or anything, and it sounds like they are willing to fix it (Which is definitely a good sign, as most "bad" shops would tell you that you did something to break it, and not want to fix it unless you paid MORE money) so that is a plus. I do agree with their recommendation to only upgrade the front speakers, and the JL stealth box's are always a good factory style option. Chances are if they sell Audison and JL they aren't a 2-bit shop, but that doesn't mean their guys in the back know how to tune a processor.
Most shops will boost frequencies until it sounds good, which, come to think of it, if this is what they did, this could definitely blow a tweeter, couple that with a bad crossover frequency and an amp that wasn't set up properly, and you could easily fry any set of speakers. Sadly, MOST shops out there, even the good ones, do not set gain's properly on an amp, and I would be willing to bet most of them don't have clue 1 about tuning a processor.
Anyways, getting the JL guy to do the install isn't a horrible idea, he probably knows most of the basics, and if it's one of the guys I know down there, they really know their stuff. Go with something like what you had, JL Stealthbox and a set of C5 or C3 components up front, a 5 channel amp, and a proper processor like a Helix by Brax or Rockford Fosgate 3sixty.3.
On the down side, either the shop didn't know what they were talking about or they were just plain lying, but going from a 700 watt 5 channel amp to a 900 watt 5 channel amp will likely not make enough of a difference to even be audibly louder at all. I'm serious, for $4500, you should have a world class sounding vehicle, get a proper install done, then find someone that ACTUALLY knows how to tune a vehicle. You will be well rewarded in the end.