What to consider while proving NP-completeness?
However, if one of the coordinates has exponentially many bits with respect to the input, then is this considered as a exponential-time checking? Or it is still polynomial-time with respect to the length of the solution?


The length of the solution/certificate must be polynomial in the size of the original instance. Therefore, if the certificate/solution can be exponential in the size of the input (due to the numbers needing exponential representation) then you do not have a valid proof of membership in $NP$. The problem might be $\exists \mathbb{R}$-hard.