Yeah, the only time Classic Juggernaut's helmet has been removed is when his force field isn't up, which was quite a lot as he didn't really need it for physical confrontations.
With the basic knowledge rule, he'll be sure to keep it up the entire fight, and he'll just wear them down. Then rape them unmercifully.
There's no reason it wouldn't. In every instance it's literally stopped everything he didn't want to touch his body.
And the few times they showed it in art, you could see it anywhere between a few inches, and a foot or so away from him.
So the lasso would still hit, but it would be around his force field, and if he dropped it for an instant, the lasso would fall as it was taut around the force field, not Cain himself.
The lasso and tiara have both been shown to get past force fields, iirc. Based on who the lasso has ensnared and effected, there's really no reason to assume Juggernaut would be immune unless he's shown to have no soul or his soul is far more fortified than beings such as Ares, Hecate, Darkseid, etc.
The lasso and tiara might have gotten by other force fields, but this one was written to resist all forms of attack and make him unstoppable...so those showings mean nothing relative to Juggs force field.
Anyway, the team cannot win without BFR; Classic Juggs inevitably wins...team loses every single time.
Especially since there is no PIS here in a forum fight...