Since the French have nothing really to offer in terms of drones and weren't really going to offer us much in engine tech, I don't see the correlation. In fact if anything, SH should get a boost because of engine commonality.
The French are good in niche products and useful as a hedge but let there be no doubt, they can't offer us anywhere close to what the US can, both militarily and diplomatically. That doesn't mean the US will necessarily offer those things but the US and the French are not on the same plane.
The IN itself is caught between 2 diametrically opposing school of thoughts with one encouraging further engagement with the US & the other cautioning against deepening the engagement & consequently , dependence. The latter comprises essentially the now rapidly losing steam Russian school of thought - IN guys.
The US for it's part much before recent developments about the ToT of the GE TF & other similar items on the agenda like local mfg of the LM Marine Engines were pushing GoI for greater commitment to the Quad especially the IN to act in close consultation & concert with the US & it's allies. MoD & GoI were similarly on the horns of a dilemma much like IN as described above.
Recent reports that the Rafale Marine was technically qualified by the IN put paid to any such tilt towards the US in this regard assuming the news was genuine. Now who knows ?
Prima facie being engaged with the US means great amount of intrusion , infiltration etc . For instance , the two Sea Guardians the IN's operating is actually being operated by USN personnel from IN bases. The P-8i base at INS Rajali in Arakkonam TN has a DoD team stationed there 24x7 to ensure compliance with the T&A's of our IGA with the US .