First, take a deep breath and don't panic. It's not that hard for Croats, yet.
If I were you, I would have registered as self-employed (no restrictions there for Croatian nationals), applied for a National Insturance number and started working... anything really. You can do certain jobs as self-employed.
As for staying legally, nobody will remove you from the country until we eventually leave the EU (unless you commit a crime or sleep rough) but you won't be treated as a qualified person. So long story short, you could stay but your time here won't count towards anything (Permanent Residence, etc.), you won't be able to use the NHS and will have no access to welfare, which is fair.
You don't need and can't obtain a visa as - simply put - under current rules EEA nationals are not subject to the UK immigration system. Croatian nationals are subject to transitional controls but there aren't restrictions for becoming self-employed.
Marriage won't be easy, as you've already learned the Register Office will look at your circumstances with suspicion
It's become tough, sadly. UK authorities in general (not only the registry) look at ID cards differently, it's not like on the Continent where ID=passport. Even your bank will ideally want to see your passport before opening an account for you. I'd go to the Croatian embassy in London and get a passport, this will make your life easier too.
Long story short, try and be creative. Register as self-employed (Link 1: https://www.gov.uk/working-for-yourself
, Link 2: https://www.gov.uk/set-up-sole-trader
). See what you can do, you could even try translations if your English is good, find a job as a sub-contractor, if you drive you could become a Deliveroo driver, anything. If you really want to stay here with your partner, you'll have to be creative.
Don't worry for staying here 'legally' (for the time being). When they say 'legally' to EEA nationals it means exercising Treaty Rights. There are thousands, perhaps tens of thousands of EEA nationals who live here and don't exercise Treaty Rights. The only thing that makes you different as a Croat is the fact that you have no access to the labour market. If you become self-employed or exercise treaty right as a self-sufficient individual, you'll be absolutely fine. Get your Croatian EHIC sorted too, this will help you maintain self-sufficient status.
Best of luck!