No, this does not work; we need analytic, not just meromorphic, continuation. If meromorphic continuation were enough, then this we would know modularity of elliptic curves in a great deal more generality than we do now.

As a consequence of Taylor's "potential modularity" theorem, we know that for any totally real field $F$, and any elliptic curve $E/F$, the $L$-series $L(E/F, s)$ has meromorphic continuation and the expected functional equation (and I think this works for character twists as well). If that were enough to deduce modularity of $E$, then we'd know that elliptic curves over $F$ were modular, which we don't.

(We do now know modularity of all $E / F$ if $[F : \mathbf{Q}] \le 3$, by results of Siksek et al; but that came well after Taylor's potential-modularity theorem and required a great deal of new ideas.)

No, this does not work; we need analytic, not just meromorphic, continuation. If meromorphic continuation were enough, then we would know modularity of elliptic curves in a great deal more generality than we do now.

As a consequence of Taylor's "potential modularity" theorem, we know that for any totally real field $F$, and any elliptic curve $E/F$, the $L$-series $L(E/F, s)$ has meromorphic continuation and the expected functional equation (and I think this works for character twists as well). If that were enough to deduce modularity of $E$, then we'd know that elliptic curves over $F$ were modular, which we don't.

(We do now know modularity of all $E / F$ if $[F : \mathbf{Q}] \le 3$, by results of Siksek et al; but that came well after Taylor's potential-modularity theorem and required a great deal of new ideas.)