It is only a 10' span and you don't have a 2nd story above so you should be able to get by with the 10". Worst case the beam just ends up thicker.
Instead of a single member PSL you just get 2 and bolt them together.
You'll need an engineer to do a quick design but really that should cost $500 and then maybe $250 for site visit after install. Some people worry that hiring an engineer will require the city but they are typically happy to design and sign off and leave you to getting permits or choosing not to.
I have a 10' opening in exterior wall with a 2nd story above - the roof load goes to the other walls though. My beam is 3.5"x 11.78" with 11' span but that is just to match my ceiling assembly for the 2nd story floor.