I suppose there are a number of ways to play this:
1. Reasoning: Sidari has to convince Bronn that taking her along really is the safest path forward, but that will take some effort. Bronn is pretty sure that the delve they're on will not be easy, and should be quite dangerous. Bronn is no simpleton, but he's also not the strongest arguer, and a convincingly presented argument can move him (that would be using some social skill, I'd say).
2. Pleading: different social skill. Likely won't work, because it will make Sidari look even more vulnerable in Bronn's eyes.
3. Something happens right now that convinces Bronn of Sidari's competence: a bar fight breaks out in which Sidari handles herself well, or some other encounter really highlights Sidari's value, as well as her ability to handle danger.