Let $(\varphi_i)_{i \in I}$ be a frame for $\mathcal H$, $T: \mathcal H \to \ell_2(I)$ its analysis operator and $(e_i)_{i \in I}$ the canonical base of $\ell_2(I)$. I want to show that, if $(\psi_i)_{i \in I}$ is an alternative dual frame for $(\varphi_i)_{i \in I}$, then there exists a bounded left-inverse $V$ for $T$ with $\psi_i = Ve_i$ for all $i \in I$. Note that the opposite is also true.
First one can see that $T$ is injective. Otherwise, since $T$ is linear $\exists x,y \in \mathcal H$ with $x \neq y$ and $T(x-y) = (\langle x-y, \varphi_i \rangle)_{i \in I} = 0$. But that contradicts the fact that $\sum_{i \in I} {|\langle x-y, \varphi_i \rangle|}² \ge A ||x-y||$ for some A, since $(\varphi_i)_{i \in I}$ is a frame. By basic function theory, $\exists$ a left-inverse of $T$ that is unique on $\operatorname{ran}(T)$.
Next, let $S$ be the frame operator wrt $(\varphi_i)_{i \in I}$. Then, $||V(\langle f, \varphi_i \rangle)_{i \in I}||_{\mathcal H} = ||f||_{\mathcal H} = ||\sum_{i \in I} \langle f, \varphi_i \rangle S^{-1}\varphi_i||_{\mathcal H} \le C ||(\langle f, \varphi_i \rangle)_{i \in I}||_{\ell_2}$, since the synthesis operator wrt $(S^{-1}\varphi_i)_{i \in I}$ is bounded.
Until this point I haven't used $(\psi_i)_{i \in I}$ but I do have a bounded left-inverse of $T$. I just don't see how alternative dual frame affects my left-inverse because again it should be unique on $\operatorname{ran}(T)$ and certainly $e_i$ is in the range of $T$.
It is my understanding, that (using your notation) a sequence $(\psi_i)_{i\in I} \subset \mathcal{H}$ is called dual frame to a frame $(\varphi_i)_{i\in I}\subset \mathcal{H}$ if it is a frame and $$ x = \sum_{i\in I} \langle x, \varphi_i\rangle \psi_i = \sum_{i\in I} \langle x, \psi_i\rangle \varphi_i $$ holds for all $x \in \mathcal{H}$. I hope you had the same definition in mind otherwise my arguments might be going in the wrong direction.
A direct consequence of this definition is that the map $$ T_{\psi}^*: \ell_2 \to \mathcal{H}, (c_i)_{i\in \mathbb{N}} \mapsto \sum_{i\in I} c_i \psi_i $$ is a left inverse of T. The boundedness of $T_{\psi}^*$ follows by the frame property of $(\psi_i)_{i\in I}$.
Apart from that, you seem to raise the question whether this left inverse has to be unique. I guess the key to that question is that you only show uniqueness on $\mathrm{ran}(T)$, but $\mathrm{ran}(T)$ does not have to be equal to $\ell^2(I)$.
Added later: You might want to check out Section 5.6 in Ole Christensen's Book: Frames and Riesz Bases, which, I think, covers everything you need to know about dual frames and an even more comprehensive answer to your question. In particular, Lemma 5.6.4 characterizes all possible bounded left inverses.