I think it would be OK in everyday communication - the "but also" can be inferred by the listener. But I think "but" alone or"but...also" would tie the two thoughts together and help the listener better understand the situation. In fact, the grammar bot is going to insist on the "but" as a conjunction to follow the comma. Otherwise, it looks like a "comma splice". But that's a rule for standard written English.
"Not only will I not get that promotion, but I could get fired." - that seems enough to get the idea across. This version seems like the best way to express things.
"Not only will I not get that promotion, but I could also get fired." Adding the "also" really emphasizes how dire the situation - no promotion AND no more job! Note the placement of the "also". I think this is the most natural-sounding place for it to go.