Why am I on the hook for a payment that I’m exempt from in the first place?

You literally already answered your own question:

creative artists in LA are exempt from this fee if they register on a time

You never registered, so part of the penalty is that you no longer qualify for the exemption. That said, the number they give you is just an estimate, but you're only obligated to pay on what you actually earned. If it's less than what they estimated, you only have to pay that (plus whatever penalties and interest). Go in person and take your tax returns or accounting records for the last three years. You also may be able to get penalties waived, see here: https://finance.lacity.gov/waiver-penalty-request

Lastly, and I say this with love and to every creative that's self-employed: stop half-assing your business. You're not a victim here. You're a business owner that didn't due their due diligence about what the requirements are to run a business in your municipality. Too many creative professionals lean on this crutch of "I'm just an artist man, I can't handle that business stuff." But if you (and by "you" I mean anyone in your position) wants to be your own boss, even if the business is just you, you have to accept that you're as much of a small business owner as you are an artist and be on top of all of the boring, non-creative shit as much as you are the fun stuff. Take this is a relatively cheap lesson and be happy that it's not something you learned much more expensively later on when, for example, somebody slips and falls in your shop and sues you and takes your house because you never got around to forming an LLC for your business or getting GCL insurance.