I was wondering the same thing. I also have plans to implement a swappable banjo with a gear set that has higher tooth count to enable a smoother finish.
I've been playing around with some gear creation software and have printed out drawings for various sized toothed gears that came with my lathe. I'm getting very accurate modeling using a module of 1, Addendum multiplier of 1 and a Dedendum multiplier of 1.
Will have to look at your calcs to see if I can improve.