After studying and researching the award a bit more, I've come to the conclusion that it would probably be better if one team nominates him for the Woodie Flowers award.
If both teams did, it would greatly confuse the judges choosing the award winner, as there would be nothing for to make them would make pick "1884's" coach over "1797's" (for example). They'd be the same person with the same assets.
(Plus, I'm not too sure how sensitive FIRST is about the same coach been nominated twice. They might view it as someone trying to "cheat", or they might not care and understand it completely. But it's always wisest to play it safe and assume the worst: that they'd view it as cheating, in which case, once again, it'd be wisest to go with only one team nominating your coach.)
Good luck you guys!
Hope I helped! (But if you have any more questions concerning the matter, feel free to PM me. I'd be happy to help you further!)