At the time of George Louis' ascension to the throne of the Kingdom of Great Britain, neither England, Scotland, nor Great Britain had ever had a king called "George". There were earlier George's in the royal families (at least one—George of Bedford), but the most recent "George" associated in any way with England was Anne's husband, George of Denmark.

Therefore, is George Louis known to have considered using a different regnal name in his new kingdom that would have aligned him more with English and Scottish traditions?

There was some (historic) precedence for this in Scotland:

When John, Earl of Carrick ascended the Scottish throne in 1390, it was deemed imprudent for him to take the regnal name of "John II", as recent kings named John had turned out badly: in England as well as in Scotland. Furthermore, royal propaganda of the time held that John Balliol had not been a legitimate king of Scotland, making the new king's regnal number also a tricky issue. To avoid these problems, John took the regnal name of Robert III, honouring his father and great-grandfather.

Admittedly, George's descent would have suggested 'Charles' to him (if the above logic had been continued) which wouldn't have been very good, but 'James' and 'Henry' should also have been available options.

With regards to using different regnal names in different places, it seems to have been possible—though perhaps not common—with Wikipedia noting Otto III of Bavaria who ruled in Hungary as Béla V.

At the time of George Louis' ascension to the throne of the Kingdom of Great Britain, neither England, Scotland, nor Great Britain had ever had a king called "George". There were earlier George's in the royal families (at least two—George, Duke of Bedford and George, Duke of Clarence), but the most recent "George" associated in any way with England was Anne's husband, George of Denmark.

Therefore, is George Louis known to have considered using a different regnal name in his new kingdom that would have aligned him more with English and Scottish traditions?

There was some (historic) precedence for this in Scotland:

When John, Earl of Carrick ascended the Scottish throne in 1390, it was deemed imprudent for him to take the regnal name of "John II", as recent kings named John had turned out badly: in England as well as in Scotland. Furthermore, royal propaganda of the time held that John Balliol had not been a legitimate king of Scotland, making the new king's regnal number also a tricky issue. To avoid these problems, John took the regnal name of Robert III, honouring his father and great-grandfather.

Admittedly, George's descent would have suggested 'Charles' to him (if the above logic had been continued) which wouldn't have been very good, but 'James' and 'Henry' should also have been available options.

With regards to using different regnal names in different places, it seems to have been possible—though perhaps not common—with Wikipedia noting Otto III of Bavaria who ruled in Hungary as Béla V.