DY, in Tana D’bei Eliyahu he describes himself as being from Shevet Gad. So according to himself he wasn’t saying that.
Also, as a side note, when he mentions Eliyahu in a Pasuk he never says, “that’s me”, or any first person reference.
Reminds of the famous question asked by a brilliant talmid chochom who nebach snapped.
Why, according to the shittah that “vaHashem berach es Avraham bakol” means a daughter, didn’t Yitchak marry her?
Answer: He was choshesh for the other shittah.