A number of people have reported issues with Mountain Lion and FMP 11 that were resolved by removing a plug in that did not work with Mountain Lion. If you have plug ins, remove them all and see if that fixes the problem. If it does, put them back one at a time to figure which one(s) cause the issue. Then contact the plug in developer to see if they have an update that fixes the issue.
Downloaded and tried to reinstall FileMaker Pro 11.0v4 Updates and it says it can't find a valid copy of Filemaker 11. I am fully legitimate.
Did you have spotlight enabled? The installer uses that tool to search for the application file in order to update it.
I turned off all plug-ins, still cannot select buttons. I created a new login and FMPA works as expected. I hid my ~/Library and copied over the ~/Library of the new user, corrected ownership, presumably resetting all preferences, and FMPA still misbehaves. How's this even possible??
I created another new OS user. Replaced all of its root directories with mine, changed ownership, FMPA works. Deleted my account and renamed the new account to my account name using Apple method. Everything but FMPA works. Renamed it back, FMPA works. So, somewhere outside of /Users 'sbradley' has a configuration file that messes up FileMaker. I just won't use it anymore.
Try deleting the preferences file for FileMaker. Not being a current Mac user, I forget the exact steps for this, but the preference file may include a link to your user account by user name so a new prefs file may resolve the issue.