Rodieke says:

FW-190A9 numbers can be tricky. The documents I have show serial production began in September of 1944.
However, a conversion program began in May '44 to the BMW 801TH motor. The aircraft will be redesignated FW-190A9. Aircraft equipped with the Hybrid Motor BMW 801TU will remain FW-190A8.
In July '44 conversions began to the BMW801TS motor as well. It is unclear whether these aircraft were redesignated. I doubt it as the FW-190A8/R11 was serial produced with only the BMW801TS motor and the FW-190A8/R7 & R8 had priority for this motor. Neither Rustsatz's was redesignated. FW-190A9's were serial produced with this motor as well.
The FW-190A9 begins showing up in the JG's in June '44.
Between serial production and conversion programs, by October '44, the FW-190A9 or it's equivalent made up the bulk of the FW-190A's.
All the best,
Crumpp