Brown's Brigade indeed had Hardee flags at Donelson.
What these troops, and many others, carried at Chickamauga is problematical. This is the biggest flags mess for the Confederates of the entire war in that the troops that fought there came from commands all over the place. There were Polk flags ,Hardee flags, ANV flags, Charleston Depot flags, First Nationals, special flags (28th Alabama), Deas Brigade flags - and perhaps the flags of Breckinridge's Division which were issued to them in late Spring, 1862.
These flags were wuite large, blue field with a red Latin cross adorned by 13 white stars. Mistakenly called the "Orphan Brigade" flags, they were, in fact, issued to his whole division which had been the Reserve Corps at Shiloh (where they used First Nationals). The other three corps had distinctive colors so Breckinridge adopted them as soon as he could.
One of the problems is what flags were the Ft. donelson exchanged troops carrying and sadly I have not located a single reference that descibes them. Maybe they were of the Breckinridge pattern; maybe more First Nationals (much of the Vicksburg garrison used them); maybe they adopted back the Hardee flags when they fought in that corps again.
It is simply only speculation that we can come up with for them.