I like Flamboyant beach villas but it has been 3 years since I stayed there.....basically right next door to the GCBC. You can walk the beach and be at the sunset cafe in about 2 minutes. You could rent two single bed units on the water if you are lucky with your dates or there is also a two bedroom unit that is not on the water. It is nothing fancy but you did say you were on a budget:

https://www.vrbo.com/346111

I believe the Bayview Beach houses are next door (between the GCBC and Flamboyant). The property didn't look like much 3 years ago but do look lovely online so I'm assuming they were renovated.