I believe you could donate blood at the red cross and they may provide paperwork to you about your antibodies. I suppose that might somehow get your test records “into the system”, but it could be a way to go outside the core US healthcare industry.
That's precisely the reason that I won't be giving blood any time in the foreseeable future. I prefer not to have my name in any database related to Covid test results.
There are IgG IgM tests for $25 at Kroger’s pharmacy, and T-cell tests; look up on T-DETECT for locations.
IIRC t cell tests can tell if one had the virus longer ago than the 6 months or so that the IgG IgM tests do.