“If someone didn’t have a hat for davening, would the Mishna Berura hold that he should put on another yarmulke? Or cover his head with his shirt sleeve?”
Most people who are worried about what the Mishna Berura has to say wear double layered yarmulkes, so actually no. Worrying about looking respectable during davening is a halachah. Having 2 layers is a minhag; so I would imagine nobody would actually recommend having your arm over your head for the whole shmoneh esreh.