Well, GeekGuy; you got me going on the silicone and you've just raised another of my pet peeves! You might prefer, to spare your feelings, to take some 'advice' from someone else here.
I don't consider a Bimini more important than sail trim. Go figure! I know some people do. To me the mainsail is the driving wheel of the boat, providing the most actual power. It's not call the MAIN sail for nothing. Mainsail trim is vital to sailing well, especially quickly. And what is sailing speed but a safety factor anyway? 'The less time you're out in it, the less can hit you.' --my dad. Fast boats are safe boats.
I would go back to the Bimini-installing guy and tell him the problem. You should have, with sail up and drawing, outhaul snug, topping lift slightly loose, about 4-6" under the boom. Cinch the topping lift, lower the sail and maybe you have 2-3". This won't hit the canvas even in gnarly weather.
You might read Don Guillette's recipe for good mainsail trim. If you haven't contemplated this before, it'll open your eyes. You'll realize leaning off to the side and ducking now and then may be worth it.
No-one ever claimed you have to have 6'5" headroom under a Bimini. Even tall people get used to it. Rarely (maybe only at dock parties) does anyone actually stand straight up in a cockpit under sail anyway. I don't have one; but a friend of mine does and on his boat I get used to standing with my head brushing against the underside of it. He has a 26-ft boat; that's to be expected. You have a 37' boat. If the boom cleared your head by 6-8" without the Bimini, there's no reason you can't have one fit without raising the boom.
Now other people will weigh in with anecdotal 'evidence' that when they raised the boom and clipped 6-8" off the sail there was 'no difference in sailing characteristics'. But there was, most likely due to the loss of draft and general area. Their boats heel sooner and farther now; and they are slower and more tender. The facts are so; math is inescapable. People's perception is that they don't notice it.
Your boat was designed to be as it is. Modifications to rigs made by non-designers almost always have detriments. But it is, of course, your boat. You pays your money and you takes your choice.
