I'm considering going up from 15a fuse to 20a fuse. Are there any perils involved? My battery rates 14a so...maybe I shouldn't?
No, you shouldn't. You can probably get away with it for a short while before stuff starts melting, but you're not solving the issue. The issue is that you have something that is drawing too many amps - very likely a short somewhere.
It's not very likely to be your little mower plug, but who knows. You should certainly begin with the right plug. I like the NGK B9ES, which I do believe is a pretty standard plug for this bike. There is a chance it can produce some radio interference that will mess with your TCI*, but you should be alright as long as you have a metal plug cap. If your cap has been replaced with a rubber one, go for the BR9ES, which has a resistor in it and will reduce the interference. The non-resistor types have a bit stronger spark since there isn't that electrical loss, but not enough to really be able to tell. The iridiums seem like a waste of moolah to me since you're probably gonna be tweaking this bike on a regular basis.
What your mower plug very likely is is too short. That means it is probably recessed in the plug hole which can cause hot spots and misfires and carbon buildup. Eww. Get it outta there!
Okay, I really gotta go to bed. Goodnight!
*Edit: TCI - not TDI. Was too late last night. I don't think there's any turbo diesel to concern yourself with!