Pretty hard to convince him, Baldwin has ten years left to live and, while he doesn't of course know this, he is, to use modern colloquialism, 'getting too old for this shit'. The stress of the abdication crisis was really the straw that broke the camel's back - he'd been in high office on and off since the Great War and the man was 70, let's not forget.
If he does stay in the job, I can see him resigning the next time there's a stressful event the government has to face - Chamberlain would probably go from PM-in-waiting to de-facto PM if Baldwin was somehow convinced to stay on as some kind of figurehead.
In short, if he stays he'll only be around for a short while longer, unless you handwave him a body and mind untouched by twenty years of stress at the highest level of government. Foreign policy (ie Appeasement) wouldn't change that much IIRC, it was on his watch that the limp-wristedness of the 1930s began, don't forget (no response to re-entering the Rhineland, and of course the shameful Hoare-Laval Pact over Abyssinia). Economics I don't see much changing either, Chamberlain was his Chancellor after all so the policies he pursued as PM would have continued.