I'll suggest you to wait till the end of July before submitting your application without the FBI PCC. As per the recent timelines, you can expect your PCC by the end of June.
Thank you both for your help. Yes, I would not want to risk anything at this point, and I hope I get my document without any issues by then, hopefully. It is just that given you have all other documents, it does get a bit frustrating having to wait for one singular paper. But I understand both of you and I will wait instead.