Well Hannover is not a big trouble: George III loved Hannover? Was indifferent to it? Hated Hannover? The problem is who his younger brothers who married both married commoners (of lower rank than Sarah) so we have two solution
a) George III was able to get the Imperial diet to convalidate his wedding to Sarah recognizing the right to inherit Hannover to their children
b) his next brother Edward, Duke of York and Albany survive and marry a Hochadel wife inheriting Hannover after George III’s death
Actually George III was very interested in Hanover and longed to visit his family's hereditary lands. He just never got the opportune moment to do so. As for the marriage itself, the Imperial Diet would play no role in recognizing or not recognizing the marriage. That was entirely up to the Emperor, first Franz I, then Josef II. So its like this: is it in Austria's best interest to have a good relationship with London or not. I think the answer is pretty obvious.