I would not have a problem going to 6.3 A, but I think a 5A SB would work fine.
If your using a SCR/TRIAC controller, I would re-think it. In that case I would use a standard blow fuse at probably 1.5x the min load and use a very large I2t fuse like 25 A. I worked with our own custom tantalum resistive heaters in a vacuum and by using controllers that had current limit and a large I2t fuse (25 A) an 3A fuse on a 120 VAC, max 200 W load. Usually the 3A fuse blew first and we saved $30 a fuse. I made sure that the current limit was in place and a fuse was rated for the load.