Oxford no longer score GCSEs anyway, although I suspect the CTEC is an issue for most medical schools? Given you are a carer though this might change the picture though as that often gets factored as a contextual flag - not sure how Oxford specifically manage contextual applicants though.
You really would be better off with literally any traditional A-level rather than the CTEC I think though - geography, music, English language, philosophy, anything really...might be worth asking your school if you can drop the EPQ (as it's rarely a factor in medicine admissions) and do an A-level in a year. Or check carefully at all medical schools to see if they accept a CTEC as a third subject.
Also note, Oxford medicine offers no specific long term benefits over any other medical school, as all medical schools are considered equal by the GMC and NHS (and as a result, specialty recruitment is blinded to your medical school to ensure there is no bias). Therefore there is no particular benefit in focusing on Oxford over any other medical school unless you present a very strong profile for their specific shortlisting criteria.