I have a similar situation. I have my First and Last name interchanged in the IRCC record and Instead of AA as first name BB as last name, I have AA as last name and BB as my first name in government record. I have been to service Canada twice and they cannot do anything apart from advising me to contact IRCC and they also advised me that legal name change request is not needed in my case. All my other IDs like driver license, health card and passport has correct order of my name despite my Canadian citizen certificate came out as wrong first and last name in position according to IRCC record. But I couldn't get a hold of IRCC and I'm lost as to what to do with that. I applied for my student loan and now my application is on hold because of the wrong order of name on my SIN . Please advise if you have any knowledge on this matter.
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.)