Frank, congratulations first and foremost.
The copay thing sounds super sketchy. Maybe the doc doesn’t want you switching to someone else later in the term? I can’t speak for your policy, but on mine there was exactly 1 copay–the first visit where the Doctor determined we were in fact pregnant. After that it was preventative pregnancy screenings and standard pregnancy appts all covered with $0 copay. We also had a $0 bill for Labor&Delivery, so that may not be the case for all plans.
Your general question about a FSA reimbursement is correct, you can pay it and then get a check from most FSA’s, just double check with yours.
A real shady move is to consider if you wife is going to stop working after the baby is born go with her company for a FSA and max contribution it. Since she’ll probably leave around Feb you’ll pay 1/6th of the total and have access to it all until she quits. If not, don’t over-estimte costs and get stuck with a ton of FSA money at the end of the year. It is use it or lose it.
As for the things your kid will need–you’ll probably have a shower and get a ton of crap you will use one or two times. Avoid buying anything until you can make sure your kid actually likes it (Zon’s swing is a good exmaple).
People will tell you “Oh this was a lifesaver for us, you have to get one” only to find out your kid hates it, and throws up whenever they use it. Blaze your own trail. Hand-me-downs and garage sales are awesome. Ebay “Clothing lots” if you don’t have access to hand-me-downs and buy someone’s entire set of cloths for a couple hundred bucks.
You need far fewer clothes than you think you do because you’ll be washing things every 1-2 days anyway due to spit ups, etc. And then when they get a little older they’ll tell you that they don’t want to wear anything except their favorite few outfits.