the good news is, you have options.... the bad news is, neither one is going to be fun.
I think routing around the bolt may be your best option... you NEED to be able to get a bite on it with something, then patience and penetrating oil, working it a very little bit at a time until you wear out the corrosion...
there is another cleaner, less invasive option, but you may not have the skills or resources to do it.... but you could have it done by an expert welder, (NOT a hobbist who thinks he is a welder)...
there is a welding rod called MG 600....after protecting the deck with a plywood template that has a hole cut to access the broken bolt, you place a very small pipe around the broken bolt to contain the weld deposit.
then you stick the MG600 electrode in the hole contacting the broken bolt.... it will weld a new bolt shank with a tensile strength approaching 180,000psi... it happens quickly so you have to work quickly.
it will heat the existing bolt enough that it will free the threads from the corrosion allowing you to unscrew it by gripping the newly added extension... ive done this procedure many times, and its not something beginners should try on something as delicate as what you have there...