I've been to FP on a budget - it can be done.
In Bora Bora we stayed at Chez Nono but another good budget bet is Pension Pemanutata. Slightly more expensive but still reasonably priced is the Maitai Polynesia.
One of the locals told us that the 4 F's are very reasonably priced in FP. fish, fruit, flowers and french bread. That was very true.
At Chez Nono they have a communal kitchen where you can self cater. We had a few take out meals from the mobile food vans over there and they were quite tasty and budget friendly.
If you want to do the motu picnic - don't do a private one with just you & DH - they have great group ones for a fraction of the cost.
HTH.