Since no one else has replied thus far, I will attempt to provide a little insight into my FBI scan experience. Warning, I won't be able to provide all the information that you're looking for but hey, it's something.
I'm applying for PR under the EE program as a FSW. I am in the final steps of finalizing my profile creation for my wife and myself. I do not have a job offer and estimate that I will receive approx. 440 points with my IELTS, degree, and foreign work experience (using the online assessment tool). With this in mind, I am hoping to take the OINP approach, assuming that I receive a LOI from Ontario. I sent in for my police scan months ago and have received my letter from the FBI. The process took exactly 13 weeks (of the 13-15 week estimate). Because I received my police scan so early in the process, I will probably re-submit for another at about June (to cover the scenario where I receive an ITA later than the 6 month limit for a police scan).
Anyway, if I was in your situation, I'd consider the following (disclaimer: this is based on research and not experience):
1. If you are waiting for the FBI scan results in the USA, it will most likely take at least 13 weeks before you will receive your results.
2. You are asking if you should go ahead and apply and attach your police scan results later, correct? I'm assuming this means that you received an ITA from the CIC? Since you have an AINP, that means you will receive an additional 600 points (I believe) on to your current score. I can deduce from the fact that your received a LOI and eventually a AINP letter that you have at least 400 points already; this would give you 1000 points, which would HIGHLY increase the chances of you getting through the application process before you receive your letter back from the FBI.
3. The CIC's website reads as follows: "Once you complete your application and submit it online, we will process it quickly. We plan to process most applications in six months or less.". It won't take 6 months for the FBI scan results to return; however, the same site also reads, "You will have to include copies of all documents we ask you for, like police certificates, before we will process your application. To keep your application from being rejected, you may want to get some of these documents ready while you are in the Express Entry pool." and "If you are invited, you will have 60 days to apply online for permanent residence.""
[CIC WEBSITE]/immigrate/skilled/profile.asp
I think the info in #3 is the most important. Keep in mind, this information is for the FSW program. My understanding of what's on the website is that you MUST apply within 60 days of receiving an ITA. Keeping that in mind, you need to ask yourself whether or not you will receive your FBI results within 60 days that you received your ITA. My understand is also that you MUST provide your FBI scan results before they will even process your application and so it's quite pointless to submit it before you have that portion of the application (the FBI results).
I hope this helps some...
