Nay. I expect Alice to evolve next update, and the MC also evolving would take away from her moment. He'd be hitting level 4, she'd be hitting level 3, he'd just overshadow her too much in terms of power. It would make anything she did fresh off the evolution seem less impressive. Maybe it could work if they were in separate fights like when they both evolved at the Battle of Diamonds, but since they're in the same one, I'd give the scene to Alice. She just lost her closest friend and has a monster with a vendetta attacking her, this is a good time for her to get to power up and kick ass.
At best, the MC might evolve next update in a later scene after this fight with Daione Danu is resolved. Probably not in combat, going straight to another big fight would just feel tedious. Some sort of emotional revelation that pushes him to evolve from mental stress, maybe? But then again, reliving his mother's death didn't do that, so I don't know what could do it now. So yeah, I'm just going to say he won't evolve next update.