What is another word for "boss" for a company with a flatter hierarchy?

I would suggest project manager or project leader. They suggest that this person is in charge of the work being done, not necessarily the people doing it (except with regards to the actual tasks involved in the project).

EDIT:
These terms could also suggest that you and your "boss" are actually complete equals, but that your "boss" is taking the lead on this particular project (whereas you are in charge of other ventures, where this "boss" would submit to your authority on related matters).

What about coordinator (or project coordinator)?

It seems to encompass both the "takes care of needs" job description and the person's ability to hire/fire.

EDIT: I don't think my answer here is correct; I'm leaving it because some people voted for it.

It sounds like you really don't want to call this person your boss, manager, lead, or supervisor. With this in mind, might I suggest Senior Engineer/Project Manager/etc.? This phrasing does not specifically state that the person is your superior in a hierarchy, but it does suggest seniority. I think whomever you're meeting will understand your meaning when you say "I'll be bringing the Senior Developer on the project with me to the meeting."

"Team Lead" has the the advantage of communicating legitimate authority while also specifying that the leader is part of the team that he leads.