What you describe is one of the many sleazy scammy practices that sadly, is WAY too common on the web. In the hardware world when companies like Apple... and ... well, Apple... do this, it's called "Vendor Lock-in" -- the intent is much the same, to 'trap' you into only dealing with them for everything. The laugh being those doing it often sell it to people as a "feature".
Did you have a contract with this developer? You know, paper, signed, notorized? - It's actually one of the litmus tests I have for how legitimate a client or developer is; if they balk at having a signed paper contract before work starts, they're probably not worth even TRYING to deal with. If they balk at having it notorized, it calls into question their legitimacy (at least if dealing within US borders).
Usually if they've locked you in like this, and you don't have that piece of paper, your options are limited. Starruman's suggestions are the way to go, and this "designer's" reaction to making what ARE reasonable requests will tell you just how big of a scam artist they are.
Now, that said, if they've scammed you on this stuff, they may have scammed you on the site itself; Even if you get domain control what they've done for a site may not actually be viable as a REAL website. I say this because the moment I hear "designer" I knee-jerk into thinking of PSD jockeys who while talented at drawing pretty pictures, they generally don't know enough about HTML, CSS or accessibility to be "designing" a blasted thing for the Internet.
You may want to have an independent third party review the site to see what else is wrong with it, and just how badly this "designer" put the saddle on you to take you for a ride.
That may sound harsh and be depressing, but there are a LOT of sleazy scam artists out there pissing out sites any old way and draining the wallets of anyone who gets duped into getting serviced by them. It's also why places like these forums exist, so we can help you get out of the hole that -- to be frank, it sounds like this "designer" dug for you, tossed you into, and filled back in.
Though I may be overreacting -- it's based on that before I retired I spent most of the last decade being the one called in to fix problems like this, cleaning up other people's messes.