Considering Constantinople had no qualms sending the Goths to Italy to crush the Italian army and its leader, I doubt it would be different with the Gallo-Romans. The East could either send an army to Italy (if this one is ruled by the so-called Gallic empire) or simply engage in diplomacy with the local powers neighbouring the Gauls (Goths, Burgundians, whoever is in Italy). Constantinople can’t really afford to send the wrong message, no one can simply claim the purple and walk away with it.
On the other end, there is ample room for cooperation between the two, as long as the Gauls realize and accept their place. Really It is not worth it to endanger relations with Constantinople just to claim the imperial title. Lots of rulers during the V century managed to legitimize their power just fine even without the purple:
- All barbarian kings had no problem ruling over their roman population, as long as their rule was framed in a more “roman context”;
- Aegidius never claimed the imperial title, he just took advantage of Gallic enmity towards the Italic aristocracy to cement his power against Ricimer and Severus;
- Ricimer effectively ruled Italy with the titles of Patrician and Magister Militum (and the support of the Italics), both with and without an emperor (on three different occasions);
- Odoacer ruled for a while in Italy under similar circumstances to those of Ricimer, the only difference being he took it one step beyond realizing there was no need to keep up the farce any longer.
An army and a fancy title (Consul, Patrician, Magister Militum etc.) was more than enough during the late V century.