Hello duckcalldan
Good day!
I suggest using IC and regional trains instead of the EuroCity which are not really suitable for bike transportation. Possibly the bike spaces are already taken.

Also, you could bike yourself from St. Margrethen or Romanshorn to Lindau. There is no need for a train, as the region around Bodensee offers great biking routes; a lot of people travel there for this leisure activity.
Kind regards,
Roland