How to upgrade generator socket from 20A to 30A?
Go ahead and put the 30A breaker in, along with a hold down kit
The good news is that your plan with the 30A breaker is feasible, given that you don't have a utility service, which means you aren't subject to the new 230.85 emergency disconnect requirements, which'd require an outdoor disconnect switch or main breaker.  However, you'll need a PK2MB hold-down kit for your 30A breaker in order to comply with NEC 408.36(D) (and keep someone from accidentally yanking out a live breaker during maintenance); fortunately, these are not hard to come by.
P.S. your panel is only a six-position panel, which means you'll have only one space left for breakers in it after you put the 30A breaker in, unless you resort to &quot;double-stuff&quot; breakers that is.