Seems like this is very much an edge case. If you have 2 different vaccination cards from different countries, you would have proof that you're fully vaccinated, but I can imagine some places not accepting this or the rules simply not being written to deal with this situation.

I would ask at the time of the second dose if they could write in the details of your first dose on the card. That way you would have everything on a single card which might avoid headaches later.

Also, check the eligibility rules in the place you're going to. The couple of states where I've looked into this would give it to anyone who could show a residence or work address in the state - and they wanted some evidence of this (utility bill, payslip or even just a business card).