Variac is blowing fuses
Variacs can and do blow fuses.   I have seen this a few times.   The reason is transformer inrush current which is well documented in other articles.   All variacs that I have seen in CHCH NewZealand are of toroidal construction and are large.  These factors make inrush current worse.  If you don't want to up your fuse rating then you could try a slower fuse     OR you can use a relay and a NTC which does work.    This inrush problem has also occurred on high power Audio Amps using toroidal transformers.    I have used a relay and a 50HZ lamp ballast to cure this and allow reasonable fuse ratings.