Maximum flow with maximum flow on specific edge
I would do it kind of the other way around as you suggested.

First compute a flow that saturates $(u,v)$. This can be done with the Ford--Fulkerson algorithm. Look only for augmenting paths which contain $(u,v)$ and augment the flow until the edge is saturated.

In the second step you augment further, but avoid the edge $(u,v)$ when searching for augmenting paths in the residual network. There is no reason to desaturate $(u,v)$. If we desaturate the edge with one path and saturate it with another we can get to the same flow by augmenting with the "product" path which avoids $(u,v)$.

This is not a proof but should guide you in a possible direction how to solve this.