Yes it is most probably the ballast causing the noise most inverters do not give a pure sine wave but closer to a square wave this can even cause some transformer and ballasts to heat up, maybe find out where you purchased the led tubes if they have a diagram how the tubes can be wireed directly without the ballast.
I am a bit hesitant to comment,as there may be other stuff available, I'm in CT; however I have converted a number of old Fl tube setups to run Led tubes of different makes: Osram and others.
In all cases the fitting was gutted of all its old parts including the starter, (the +- 20mm Dia x 35 mm long starter).
TheTubes were supplied with the tubes' fuse (looks like a starter) was put in where the starter was. The supplier told me this and warning stickers came with to say Led tubes only....to be stuck on the modified to Led fittings.
Led (Fl style ) tube fittings have nothing inside just wires and a connector block.