How to pronounce the adjectival form of "patent" in American English?

Solution 1:

If you are speaking of leather shoes, or of something that is obvious, then the first syllable in BrE rhymes with 'pay', as in wages.

But, if you are speaking of an official document that confers certain rights, such as the exclusive right to exploit some invention, or that formally appoints someone to an official position, then in educated BrE the first syllable of 'patent' rhymes with 'patt'.

The UK government office that licenses inventors' rights is the 'Patent Office' - rhyming with 'patt'.