Hi there! Thanks for the ask 🙂
I think James and Lily asked Sirius to be godfather before Harry was born, tbh. It was wartime, and they knew Voldemort was after them fairly early I believe (though JKR has given conflicting timelines for when they went into hiding). I think they wanted Sirius to know their intentions right away… just in case the worst happened, to any of them.
I see dogfather Sirius as being a little shit, getting Harry the latest toys that are dangerous, or sing annoying songs, or are prone to catch fire, etc. He makes jokes about taking Harry on his motorcycle and feeding him the booger flavoured Bertie Botts beans.
But when he’s actually on official babysitting duty, he panics a bit when he realizes, oh shit, I am in charge of this tiny thing, this easily breakable little poof, and- no, Harry, don’t touch that! By the time James and Lily come home, Harry is seated on the couch wearing Sirius’ motorcycle helmet, surrounded by a barricade of pillows.
James and Lily always knew they made the right choice. They always had complete faith in Sirius. Hence partly why they were confident in his secret keeper switching plan.