Insofar as it sounds like the shoulder nut is not normally removable and is necessary for the proper functioning of the French press, I believe that not only does it not require tevilah (immersion in a mikvah) the fact that the Jewish owner must preform this assembly would negate the need for tevilah for the entire French press itself [presuming the repair was required before it had been immersed already]. The circumstances seem to imply that this repair is a halachiclly significant act of completing the utensil and as such is is "Jewish made" and requires no immersion. Of course it is clearly a case where a posek should be consulted to judge how significant the assembly is.
(Furthermore, even if the act wasn't significant enough to negate the need for immersion of the French press, it is probable that the shoulder nut alone does not have a status of a utensil until it has been added to the French press, which we have noted is done by a Jew in our case so no tevillah is required because they are the one who "made" it).
A possible suggestion would be to disassemble the vessel and have a
Jew reassemble the vessel. Reassembly would not apply to the parts of
a meat grinder that are regularly assembled and disassembled during
ordinary use. Reassembly would apply to utensils that are not
An alternative option is for a Jew to take apart the appliance and re-assemble it.
Firstly, an appliance made by a Jew doesn't need to be dunked in a
mikva. If you can "break" and then "fix" the appliance yourself, that
works. (What's called "breaking" and "fixing"?)