It's actually common practice to up the protein when the molt begins in order to supply enough protein for feather regrowth, especially if they are still laying eggs. I have never heard in my life that upping their protein/nutrition sends them into a molt. Never had it happen here. If you starve her, yes, that would do it, but upping protein? No, I'm not convinced of that one. Some folks don't add protein during the molt, but if you do, it won't cause them to molt if they weren't already ready to molt. We're talking about barely more protein than normal here at just 20%. Not sure what would happen if you added way too much of it, like 24-30%, as in game starter. 20% isn't too much protein at all, especially in times of stress. If you starved them, they'd surely go into a molt in response to that.
And it is not at all uncommon for them to go through a mini-molt at 9 months old--happens with every pullet I have. The degree varies-most of them only seem to lose some head/neck feathers and a few from other areas, but last year, I had two 9-10 month old pullets lose tons of feathers plus a noticeable amount of weight.
I have one coop with my elderly hens in it. When they molt heavily (and one has severe arthritis in her feet), I add one heat bulb so they have a "heat station" that they can go under if they need it but usually only Lexie will get directly under it--her feet really hurt her. It does not heat the coop at all and that coop has 9' ceilings and is not insulated so I do that for my five year olds. Each molt seems progressively harder on them as they get up in age.