Dechavilier admits that he has no idea how to deal with the problem but What if Kerensky decides on or about this time to take the Royals out of the SLDF and reform the HAF headed by the Royal Command.
Kerensky couldn't/wouldn't do it.
The conversion of the HAF to form the core of the SLDF was written into the Star League Accords. So reforging them into anything is/was illegal. His power as Protector of the Realm was largely ceremonial, and what little political power he had was negated by the other House Lords. But honestly, seeing how Kerensky wouldn't helm the Hegemony, usurp the Council Lords, or do anything except leave the Inner Sphere to protect his precious ideal of the "Star League" this would never ever happen.
A guy like Dechavilier would suggest it, and maybe even attempt it if he were desperate enough – after all, he suggested tossing the Council Lords, etc. – but is still highly unlikely.
Try to remember that the tension exists between soldiers of the Star League, not between soldiers and the Star League. Keep the Star League out of it.
That said, changes within the bureaucracy of the Royal Command could have a positive long term effect, as would some unit rotations. Modifying the supply situation would also help, as would a round of SLDF Regular upgrades – particularly among mid-rank officers – nothing pan-SLDF.
Seeing as this is an AU, I'd make changes to Kerensky's character, or slowly siphon off troublesome Royal units to help enlarge the Hegemony militia groups. Technically, tossing Divisions to the militia isn't illegal – in fact, the militia is the only remaining bits of the HAF left in the Star League – so that's one way to quietly rebuild the HAF.
That might be a tough sell to front line troops, and it wouldn't help with the navy. Mothballing is your only answer there. Plus, you've stuck with canon dates, so there isn't much time for a slow and quiet reorganization.