Can I be honest? You first need to look within before you can look outside. When I read your words, what I see is a team member who is too frustrated with their current avenues to see all the other choices before them. Who said you had to stay within engineering? Who said only HQs give out funds? Who said the community had to be united to support something? Who said your team can't fundraise with another team and split the profits?
To start: List the companies you've already contacted, contact your local chamber of commerce (
www.culvercitychamber.com 
), obtain a list of companies in the area, make up a fundraising letter, and send them out.
Fundraising is a lot like trying to invent something: you try over and over again, and just when you feel that you've tried all the possibilities and all the doors are shut...another door opens.
Noone said fundraising was easy....some teams may look like have it easy in the fundraising area, but they've got other problems that you may or may not see...and in those areas where they have problems, your team may be better.
My other suggestion for you today is to take the initiative and personally contact teams in your same predicament: teams in a metro area all sharing the same resources. Manchester is the classic case for this predicament: somehow, four teams have shared that city since FIRST's creation. I'm sure they've been in the same place you have and seeing as though they're still standing, they must have found some solutions.