I agree with you in theory, this is what I would try to do. Now the big big problem is. I cant get fingerprints anywhere here in Germany. I called several offices (police, public service, immigration...) they all told me that they are either not authorized, nor have the capacities or the proper qualifications to do such a thing. With police here, I would literally have to commit a crime for them to take my fingerprints. Which naturally defeats the whole purpose of getting a criminal record check.
Now what is left is to hope for an extension by IRCC, or what would be ridiculous, fly out to Canada for a week an do it there. This is really annoying.