You will need to identify a calm, cool, representative of the team to negotiate with the sponsor and carefully and calmly explain a few things.
a) Your team identity belongs to the team, not to a sponsor.
b) an example of signage:
http://kellrobotics.org/files/Billboard.pdf The partners and sponsors are placed in the 'whitespace' and are listed horizontally. Placing on whitespace is a way to deal with corporate color issues. However it doesn't solve your tee shirt problem.
c) Many companies have a monochromatic version of their logo which would work on a shirt like yours, but it sound's like they don't have one.
d) The fundamental mission of FIRST and your team is to promote the mission of cultural change regarding STEM. It is NOT an advertising forum for a company, NOT a trade show, NOT a place to hand out corporate brochures or advertising, at least not directly without other consideration.
e) If they are fully bought into these concepts and are partnered with your team as a culture changer, then there shouldn't be any problem with them making reasonable adjustments. Otherwise I'd consider returning the money and finding a more appropriate 'partner'. Note I didn't say sponsor, but partner, there is a difference. They would have to be a VERY heavy hitter in order to make these demands, 15% doesn't cut it in IMHO.