I'm not so sure about it being totally against German culture. Luxembourg, which may be considered German as well, since it had been part of the HRE, the German Confederation and the Customs Union, had two female monarchs: Marie-Adelaide and Charlotte.
What led to the change in the succession law in Luxembourg, can lead to the same in Hanover. In Luxembourg, the monarch had no legitimate heir (at least that's what he said), but he did have daughters, so he convinced the legislature to amend the relevant law.
If William IV can convince the legislature that Hanover must remain in personal union with the United Kingdom, then we have the desired outcome. The question is, why would William, or any prominent British politician, even care? I think they had no problem in letting Hanover go, given the messed up situation in Germany. Prussia was already ascending and, sooner or later, it would try to form a united German state, including Hanover ofc. Plus, it seems that nobody in London thought this northern German state as too valuable; after all, it was India that was the jewel of the Empire, Hanover was not even close. (1)
The fact that William IV signed a more liberal Constitution for Hanover would, I think, increase his chances that he would get any amendment to the Hanoverian succession law passed; plus, money can go to the right pockets and everything be 'settled'. What we need here is the will to keep Hanover closely bound to the UK.
Finally, another problem to consider is this: iirc, the kingdom included many seperate realms with their own Estates. Unless one wants to seperate the core Hanoverian lands from the various other realms which together formed the Hanoverian Kingdom, one also needs to convince all these Estates to concur with the change in the succession law. Suddenly, very deep pockets are needed (and Queen Victoria does not personally has them atm).
(1) And it could never be, Hanover was not a British dominion or satellite. People in London knew they could interfere in Hanover so much.