Hey @George_Thompson, so we finally merged our data this week and tested it out on PostgreSQL (up to date) and after testing several things such as copying the old database to PostgreSQL, recreating a new table and appending it to PostgreSQL, I'm still getting this error.
In testing though it seemed to be fine initially before I created templates for the feature class I am actively editing, but that could have just been a coincidence. I haven't tested it for any great length of time to see if that is for sure part of the problem or not though.
Currently the only solution is what @MartinWong1 suggested and that's enabling editing and having it set to "single workspave edit session", which is what I had to use previously before we merged the data to PostgreSQL this week...
Error code given ArcGIS Pro with the versioned data now in PostgreSQL:
Note: Last night before I left work, things were working fine without the Enable Editing checked off. I left it over night and this morning is where I saw the error. So it seemed to occur even without me present overnight, if that is a clue or anything?