Most FSA’s i’ve had have let me self-pay and file a claim for reimbursement.
I have a 3 month old. I added them to my plan with an HSA and upped my HSA contribution and got another $500 company contribution for switching from an individual to me + kid plan (my wife likes her plan and it isn’t worth the argument).
Check the Facebook virtual garage sales for things like swings that your kid may or may not like. We bought a $250 swing for $75 that my baby used for a week and then stopped liking it. We sold it for the same amount. Also have bought other things like safety gates and a like-new backpack for carrying the baby on hikes.
We bought last year’s model of our stroller online for about $200 less than the current year.
There’s a book called Baby Bargains that my wife swears by.
Relevant thread on college saving (can’t start too early): Best way to prep for future College costs? - #4 by redmill