I have a tendency to do every sidequest in the game for the sake of completionism. However, I usually end up with a level too high for my liking - and I definitely do not feel like entering UVHM at level 55 with level 50 gear. Should I avoid specific story quests in TVHM? Maybe run through some of the areas that scale to the character’s level with as little fighting as possible?
Once you hit level 50 in TVHM, the game will automatically lower the amount of XP that you get from kills or missions. As long as you only do the vanilla game, you’ll be fine. Even if you do all the missions. You may get to level 51, but most likely you will remain at level 50. But… If you do dlc material, you need to think carefully about how you wanna enter UVHM. If you have access to gear that’s around 54-55, then NP. But if you, as you say in the OP, you only have level 50 gear, then I’d suggest doing only one dlc, at the most. UVHM is rather brutal the first time you go through it.
If you still decide you wanna do all the dlc material in TVHM, you will have to struggle a lot. One way to mitigate this is to start your UVHM journey by doing the Mercenary day headhunter pack first. That way you have a way to farm for good gear as you progress through UVHM. It will still be a rough start, but it makes things easier from there on.
I would suggest skipping side quests unless unless you feel the need to get a bit more XP (i.e. you’re actually under-levelled for the story). Work it so you hit level 50 either before or during the run to the Vault and the final boss.
Once you’ve completed main story, grab some of the better side quest rewards (Rubi, Heartbreaker, …) and start UVHM and press through to Liar’s Berg. You can then switch go back to TVHM and finish off side quests, but you’l have access to vending machines in UVHM that will now level with you (along with a farmable boss) when you come back to UVHM. You can even leave things like the Hyperion Circle of Slaughter until you’ve advanced more in UVHM (say, to Sanctuary).