If one background check is done, the other should be done as well. Call the SPBI back with your money order info and find out when they were returned.
They have '1 week' to issue or deny per CGS 29-29 once they receive the background check back, otherwise you should be appealing. I would be willing to bet that at the 5 week mark, you are overdue for an appeal, you just need to get the dates.
It is (unfortunately) very common for PDs to talk about extraordinary amounts of time despite they the fact that they effectively have about 2-3 weeks to issue or deny.