-) Why do they need to be thumbscrews? Small Allen screws should do well too.
(If you do not want to use an Allen key, you might try filing ridges into the heads.)
-) there are thumb knobs to be installed on sink-head screws, turning them into thumbscrews
-) thumb screws are available off the shelve at few shops specialized in rare screws (post #2)
-) any machine shop doing single part jobs can easily turn you a thumb screw
The size of the Horseman thread for sure some Apugger can tell you.