Just an extra bit of info to clarify. If your view is PT1=Normal, PT2=TVHM & PT2.5=UVHM, that isn’t quite right.
In BL1, the last chance you had to do the missions for specific gear was in PT2. PT2.5 just let you carry on looting. In BL2/TPS, UVHM resets the game and you get to do all the missions, again, at a higher level.
The level cap increase was applied to the base game regardless of having the DLC that gives you UVHM. If you continue in TVHM, after defeating the Sentinel, you will carry on leveelling up but painfully slowly Any missions in TVHM, no matter when you do them, will never give you better than L50 gear.
If you get the DLC and start UVHM, even the best L50 legendaries will soon become out-classed at this tougher level. If your objective is to have the best gear at the highest level, then you want to save your side missions during your UVHM playthrough.
There is a further difference between PT2.5 and UVHM. You are able to reset UVHM and do it all again, as many times as you like. You could play through UVHM to hit the level cap, reset the game and get all mission-specific gear at the maximum level.
I hope I’ve answered the right question.