Why is Lloyd so bent on Kim proving her lawyer status?
It's a meaningless distraction to divert the spotlight away from the real issue which is his poor behavior.
Like in "A few Good Men", he's hoping that the mere appearance of something wrong will undermine all claims made.
It's what he does, what he's always done - try to find something, anything, however insignificant so he can then say "ah ha, see, it's LIES, it's ALL LIES". Of course it isn't, but he's easily convinced, especially by himself.
And really, even if Kim was never ever a lawyer ... what the fuck would that have to do with the price of milk? (that's a UK expression, to highlight how one thing has no bearing on another).
But I have no reason to question anything she says and many many reasons not to believe anything that comes out of his fat hairy gob.