You can put rigify on any humanoid base mesh. There’s nothing “special” about MB-Lab that prevents it.
Make sure you aren’t applying the subdiv modifier. You don’t want that applied, you can put it back on later. I wouldn’t apply the displacement either unless it is controlled by weights, and I would finalize the character before messing with the modifiers. Also make sure you’ve applied any transforms to your mesh, and to the rigify rig before parenting.
Could help to delete the MB-Lab human’s weights from the mesh - I’m not sure how auto-weights plays with old weights. Once you’ve removed the old armature and weights, you have a static mesh with nothing special about it merely from coming from MB-Lab, if it still isn’t working, try another Rigify tutorial on youtube. (One warning about rigify, you need some of the neck bones to remain “connected” for it to work (the tail and head of the bones must remain in the same location as each other), even though the rig lets you move them independently)
To me, it just looks like a bad setup in the metarig. I’ve had those before, usually it’s a wrong roll on a bone or stupid stuff like that, which are easy to not notice until you generate the final rig.
So yeah, please upload your file with metarig, generated rig, and binding done as in your screenshots with issues.