Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, this leads to a new manifold $(M,\tilde g)$ with the same underlying differentiable manifold but with a different metric. My problem is when I get equations involving both metrics. For example, if I consider a transformation of the type
$ \tilde g = g + T, \tag{1} $
where $T$ is a tensor field (assume that it satisfies everything that has to be satisfied so that $\tilde g$ is a metric indeed). Then I can express the Ricci tensor, for example, in terms of the new metric and I get in coordinates something like
$ R_{ij} = \tilde R_{ij} + B_{ij}, \tag{2} $
where $\tilde R_{ij}$ has the same analytical form of the Ricci curvature but with the new metric $\tilde g$ instead of $g$ and $B_{ij}$ denote collectively other terms depending solely on $\tilde g$ and on $T$. Now, if I want to find $R^{ij}$, should I use $R^{ij}=g^{im}g^{jn}R_{mn}$ or $R^{ij}=\tilde g^{im}\tilde g^{jn}R_{mn}$? What about $\tilde R^{ij}$ and $B^{ij}$? It seems weird to me because the LHS of eq. (2) depends on $g$ while the RHS depends on $\tilde g$.
"Raising and lowering indices using a metric" is only a notional convention and has no real content. In a situation like the one you describe, you can either decide to consider one of the two metrics as being fixed and fix your conventions by using only this metric to raise and lower. This is consistent but potentially dangerous. For example in situation like with Ricci curvatures (which is canonical with lower indices) you get effects like that $R^i_j=g^{ij}R_{ij}$ is the scalar curvature of $g$, whereas $\tilde R^i_j=g^{ij}\tilde R_{ij}$ is not the scalar curvature of $\tilde g$ (which equals $\tilde g^{ij}\tilde R_{ij}$). But it is a convention that works if you are careful enough.
The alternative (which I would probably recommend) is to drop the convention completely and write out the rasing and lowering of indices explicitly all the time.