I also have notchy 1/2 on cold but I thought this was normal for the tremec.
I have never experienced any issues 1-2 or other gears on my 5. Cold, warm, never an issue.
The problem is notchy shifts are subjective. If you all keep shifting into 2nd and it doesn’t slip into gear as smooth or as easy as any other gear then the synchro is wearing down and eventually it will need to be replaced (partial rebuild). Anyone who experiences this should be double clutching in 2nd till the transmission fluid is fully warmed up or rev it out to shift into 3rd.
The dealer is doing the right thing by replacing it, the parts to rebuild it may not be available, they may not have a tech with that expertise and it’s still a newer car. If it was an auto they’d never risk a rebuild at a dealer so it shouldn’t be any different with a manual transmission.