On the other hand, in French it seems, that the difference could be critical.
Do not overestimate this criticality.
I have pronounced rechercherai and rechercherais exactly the same way for my whole life and do not plan to change. Many people don't even noticed it because they are doing the same and none of the remaining ones found it critical enough to ...
I do not expect any other pronunciation than /miɲɔt/ by a French person. It is close to (UK) English "min1 yacht", the middle "gn" is pronounced like the Spanish ñ, close to "canyon", the name rhymes with "pot".
Mignotte is not just a proper name but also a noun/adjective in French. Despite being relatively outdated ...
This is not exact, just two of these forms are pronounced alike "rechercheras" and "recherchera".
Futur simple : -erai, -eras, -era, -erons, -erez, -eront
Here is a source that provides the sound for the verbs in the first group and the other groups: prononciation des terminaisons au futur; this is the pronunciation in the south of ...
E caduc/schwa realization is a regional and individual choice.
In careful speech and poetry this E is generally pronounced. It is also the case by people with the southern French accent.
In all other contexts, it is most often skipped.