Obviously, fan fiction cannot be copyrighted or sold because it’s a derivative work of licensed content. I know all of the legal bits regarding the direct profit from the work itself.
But are there any other methods for generating even a little money? Is begging for Patreon or some other tip service the only option?
Once you start asking for money, you run into risk. Most of the time, publishers and other rights holders don’t mess with fanfic, but when they do, it’s almost always because someone made that mistake.
Tbh, it doesn’t even matter how much you lift in that regard. You could win the resulting lawsuit over your mouse with gloves named Brickey, but you’ve still had the hassles, and you can’t predict with full certainly which IP holders are going to be aggressive like that.
Most likely, they’d cease and desist you first, but that’s not a guarantee.
I’d just leave it as being for your kids, and if you want to share it, don’t even ask for money at all.