Should I try to ask company B again for an increase
If the salary they offer won't work for you there's no harm in negotiating. You can bring up that you got a higher offer, that's not a faux pas. I typically don't advise negotiating based on other offers but the scenario you describe is one where it's not a faux pas and bizarrely it can be more effective than arguing on proper reasons. Do not mention the signing bonus. It's largely irrelevant and risks confusing your message. Some companies give them, some don't. You can ask for one during negotiation but that you'd get one at A shouldn't be a basis for comparison.
They may balk at renegotiating and simply withdraw their offer, but I'm assuming you wouldn't take it at that salary anyway. If you would accept their offer due to other reasons (benefits, location, growth, ...) then you need to be a lot more careful in your negotiation.
Don't ever believe "we may give you a raise to X in Y months". If they make that point then you need to get that promise of a raise in writing from the hiring manager. No "we'll evaluate your salary". No "conditional on good performance". It should a black-and-white "salary increase to X after 6/12 months".
Of course, if A and B are equally interesting I wouldn't even bother with this. 15k is a pretty significant gap, even if they mentioned a number close to that as a possible raise somewhere down the line.