I'm athiest, but spent many years in church, so here's my understanding of why he references the Old testament:
Jesus said he came to fulfill the law, not to abolish it. (Matthew 5:17-18) So the Law, which is found in the OT, is still in effect. Each denomination and individual decides for themselves what parts they'll pay attention to.
Many Christians throw out the entire Old Testament, but most just pick and choose which parts they want enforced. They love the 10 commandments, for example. But they also wear mixed fabrics, eat pork & shrimp, and some get tattoos...all of which are forbidden in the OT.
This guy plays Pick & Choose. If it's in the OT and he can use it to make this country a "Biblical Republic", he will use those verses. If it means he can't eat crawfish etouffee, then that verse clearly doesn't matter to him.
The scariest part about this dude is that he interprets everything thru the lens of scripture. And he does not believe the separation of church and state is real.