MUL-T has lots of health, but he suffers from his slow speed. His Transport Mode does more damage if he is faster, so stacking speed definitely helps.
MUL-T's nailgun is high dps, Leeching Seed keeps him alive against healthy enemies.
Increases MUL-T's critical hit chance and increases his dps.
Just like Commando, MUL-T does a lot of proc damage too. These items all increases MUL-T's overall dps.
Stacks MUL-T proc damage, and the explosive damage is also a great AoE to enemies around.
Soldier's Syringe can definitely help with MUL-T Rebar Puncher, but it is unnecessary. Soldier's Syringes would be good if it is a solo run, but give them to your teammates if you are in multiplayer.