The confusing part of "This action cannot be performed because this field is not modifiable." is that Filemaker doesn't tell you which field is "this" field. You should be able to use a global field in the manner you describe. Given the error message you describe, it's possible that there is step in your script that is triggering it.
Generally, you get this message if you attempt to modify the contents of a calculation field or a field that has been set to prevent user modification in its field options.
Check your field definitions and examine your script carefully. If you are using fmp advanced, you may want to use the script debugger to step through your script.
Ok, I changed the field to not execute a script at all to see if I could get the selection to work, and I get the same error. Per your suggestion I looked at all the field definitions, and it struck me a bit odd that you sited a calculation field as a potential cause, since I could choose only a calculation or a summary for the field type. Something I should have mentioned is that this table is actually an ODBC connection to an Oracle table. I was adding the global field to this table, which FileMaker allows, but these fields are limited to just calculation and summary field types for non-native Filemaker tables. So, I created a new, regular FileMaker table and added my global field there. Then, I was able to choose text as the field type. I rigged everything back up... the error is gone and everything works. So, it was a calculation field problem.
"Something I should have mentioned is that this table is actually an ODBC connection to an Oracle table"
This was a pretty big oversight...