I have been applying with my American husband for PR in Canada. Recently, CIC had requested that we do our Canadian background checks as we have stayed in Canada for more than 6 months previously. In doing so, we have had some trouble finding a company to run our background checks as the RCMP couldn't do it. Especially not overseas. I have noticed that other people were also having the same problems as us. So I am just posting on the company we went through to help others out.
We went to the Fingerprint Room Inc. and they were able to help us get a FBI and Canadian background check super easily and quickly. All we had to do was get our fingerprints done locally and hand over all the work to them. We pretty much filled out the online application and they helped us get the RCMP certificate within 10 business days. The best part was they used tracked FedEx shipping, so our documents were safe and arrived shortly. They are also very friendly people, constantly updating us with the process. Here is their website https://thefingerprintroom.com/
I hope this helps! The time limit CIC gives us is so stressful, but they made it so much simpler. I'm glad I was able to find them.