You have several options.
1. You could apply for a place now, via Clearing, using the grades which you have (the grades you're not happy with). This might not be at you preferred university, but it's an option.
2. You could embark on a retake year and complete a new UCAS application to start uni in September 2023. Will your current school/college let you retake your A levels there? That would makes things simpler (in terms of predicted grades, a reference for your UCAS application, etc). If not, as you say, you could study independently and resit privately via a separate exam centre. Even if you can't retake via you current school/college, they are in the best position to write a reference and predict your grades (knowing your ability and circumstances). Would they be willing to do that, do you think?
3. Alternatively, you could do as above (i.e. do a retake year) and then apply with your actual grades via Clearing this time next year, to start in September 2023. (Although your only options will be those unis/courses which are available via Clearing then.)
4. You could also apply next year (with your actual grades from the retake year) to start in September 2024.
The good news, is that you can actually do all of the above:
1. Look in Clearing now for a course which might suit. (You should see a blue "Your Clearing matches" in UCAS Hub which will show you places which are available where your grades meet the requirements.)
2. If you don't find one, begin a re-take year and apply to start in September 2023 (using whatever predicted grades you'll have this time, together with your current actual grades).
3. If you don't get any offers, or your retake grades don't meet those offers, then look in Clearing this time next year.
4. If you don't find anything you like there, re-apply from next September using your "new" actual grades to start in September 2024.
If you take this approach, you don't actually have a decision to make. You just follow the above steps until you (hopefully) secure a place you're happy with.