So far as I can see, OP's example 2a could never be valid. 1a requires a very specific (slightly contrived) context to be "valid/natural", which might be easier to see if we consider...
1: As breadwinner, I don't make quite enough money to support myself (or my family) in this job.
2: Dad had some hang-ups about his working wife taking away from his role as breadwinner.
In both the above contexts (which I consider to be perfectly well-formed sentences) it would be quite acceptable (but not required) to include the definite article ("the"). That's because of the pragmatically strong implication that there's only one breadwinner in the current context.
3: Besides myself, the only other people employed by our tiny village school are the groundsman, the cook, and the part-time secretary. So as [the] teacher, I get to meet the parents of every pupil.
In (3) above, using the indefinite article ("a") would be extremely unlikely, if not actually "invalid". But as a general principle, if using the definite article would be credible, it's also possible to omit it.
Assuming OP isn't a native speaker, the "correct" answer for the most common context is Include the indefinite article. But if that's all that's being asked about, perhaps the question should have been asked on English Language Learners.