Hi Luke
I can confirm that the change of ATO for integrated training is possible and can be done. We have done it quite a few times ourselves.
It is way easier if you do it without any flight hours!!!
If you have flown some hours, you can still do it bu then the recipient ATO must check each and every flight from your previous ATO and compare it with his Integrated training... What could happen? that some of your flights cannot be credited...
So moving when there are no flights completed it is much straightforward.
The way we do it, is we would make a recommendation to our Spanish CAA (AESA) asking for approval of our suggested training in order to complete your integrated training. Just a last comment... note that the MCC is part of the integrated training... so you must complete it with the school you choose (We are aware of some authorities not checking this thoroughly but this is the regulation)
In theory, there are no credits moving from Integrated to Modular... I know it is hard to believe but it is like that. If you wanted to do that, you would need to consult with each specific ATO because the way the same EASA rules are implemented in each country are not quite the same.
All in all, I think we have between 10 and 20 students that had moved from other Integrated schools...