There are a few profile fields set to arbitrarily short lengths and these are the subject of numerous posts in moodle forums. Increasing their size is possible but involves core changes which then need to be carefully tracked. At first glance though, this may be the best option if it is absolutely impossible somehow to make your intitution names fit.

For Totara Sync to work at all, the "ID number" must be a unique identifier for each user, and it's certainly inadvisable to attempt to change this. 

There seems to be a lot of confusion around the "ID number" field. I don't believe it is actually enforced unique by the database (unlike the "id" field), but Totara Sync uses this to synchronise with external systems which will have no knowledge of moodle's id field but have their own unique id system and Totara backs this up by checking its uniqueness during synchronisation. Here Totara is assuming the external system is "master" so must use the external ids.