I believe expending fatigue for dice are restricted to an attack roll, and this is not one of those rolls, so no you may not.
You are right here
Not sure where you getting this +1 from though?
RtL Rulebook page 31
A hero who is caught in a web is marked with a web token. At the start of each of that hero’s turns, its owner must roll a black power die for each web token on it, plus one additional black power die for each die of Melee trait the hero has (the number of bonus power dice rolled when the hero makes a melee attack). For each power surge rolled, remove one web token. If any web tokens remain on the figure after rolling, the figure cannot spend any movement points that turn.
Monsters caught in a web remove tokens the same way, except that they roll one black power die for each web token, plus one extra black power die for each space they occupy beyond the first.