In theory, you can get any of the maintenance organisations with the proper approvals to do the work. All you need to do is to let your CAMO know. You could change any time without worry.
Again, in theory, there is no link between your CAMO and MO, as these should be arms-length arrangements. A lot of the larger companies have effectively split the roles within the organsation.
However, it is quite possible that if you are under a restrictive deal with the CAMO and that they will not accept you moving away from their preferred MO. I would doubt that would stand up in court though.
Your problem is that you are going to be stuffed by the CAMO and if you have signed up for a controlled environment, it will be expensive and difficult to move.
In one of the June editions of the flying mags there is a summary of Part M. The most interesting bit there concerns the new ELA1 regs (applicable to sub-1000kg, but likely to be sub-1200kg).
In it most of the low-end fleet will not have to sign up to the controlled environment, as the rules are a lighter version. These are not set in stone yet and the CAA has been delaying letting the GA owners know about this and also approving the engineers that could make it happen.
Again, a piece of nastiness from the Belgrano.
Things are likely to get a little worse soon with the new Chairman of the CAA being a chum of the Treasury and a non-flier.