Sorry, it seems your inquiry fell through a bit, despite being relevant and well written.
I would guess that the problem really is that with a small dataset, the data do not really inform both alpha and beta, but only alpha_times_beta.
If you want to dig deeper, I would recommend Mike Betancourt’s post on non-identifiability (https://betanalpha.github.io/assets/case_studies/identifiability.html) and possibly also on underdetermined regression (https://betanalpha.github.io/assets/case_studies/underdetermined_linear_regression.html) as this should give you enough background to be able to investigate yourself.
If you’ll have further questions feel free to ask!