It's not entirely clear what happened here but - IF it truly was an error by IRCC - go to your MP and ask for assistance.
Again, IF (and only if) it was their error than the advice from service canada may be wrong.
[BUT - there is something that doesn't hold together in your explanation, ie 'interchanged in the IRCC record' - was it like that from the beginning? When did the error come up? When you became a PR?]
Also, if it was like that for many years before you decided to deal with it (even if originally it was an IRCC error) - well, now it gets complicated. Use a name long enough, that's now your name.
If it is NOT an IRCC error but something else, you probalby will need to do a legal name change. YOu may also want to see a lawyer/legal assistance; it's possible there is some interim step that could help with your other issues. (There's a way to swear an affidavit that eg AA BB and BB AA are one and the same and I am them - but again, don't know if that will help in your case.)