Okay, first things first :
You'll have nothing to fear from Brexit in your case. Freedom of movement is highly likely to remain, although there might well be caps on employment. In your case, visiting Poland for 10 days a month will be covered by the tourist visa waiver if the freedom of movement stops - you'll have an entitlement to spend 90 days out of 180 in the Schengen zone.
I don't fancy applying for visas in some years time and apart from anything else I don't see what catagory of visa one would apply for.
If it came to it, you'd apply for a multiple entry long term visa. Schengen visas can be issued for up to 5 years, and in your case (a child in Poland), it would be issued pretty much without any fuss. But there's no way Poland will require visas from British people in the future.
. Register for what ?
You can register as a temporary address for up to 3 months using your rental contract. Longer than that will require proof of exactly what you're doing here, and won't be granted without evidence.