I can't say anything about Mimetex. So I won't.
What were the reasons behind your hosting service saying it wouldn't be up to the task?
Just to see if I could, I recently installed Moodle 2..5.x from my shared hosting service, Hostmonster, auto install script. And to be honest it worked a treat. I was quite surprised. I could even use git to install all my usual plugins etc.
The reality though is that Moodle is a fair bit more resource intensive than, say, WordPress. And you would want a VPS at least, and quite probably a dedicated server if you plan to have classloads of students going at it simultaneously. The hostmonster account would not be up to that. Possibly your hosting service was aware of this, or they are running a very old version of PHP.
If you have someone to look after your Moodle server, from your IT dept. Then running a server that is accessible from outside the school network is a good option. But if you don't really have that, your best bet is to find a good Moodle hosting service.
Whats a good Moodle hosting service? Well, you could maybe google that, or look here on the forums ..