Even if this is possible given his ridiculous contract, a #4 for Bradford is lunacy.
Ignoring his play entirely for a moment... if STL wants to move Bradford it's because they're taking whichever quarterback is available at #2 and Bradford immediately becomes a $60M anchor on their salary cap. If they balk at the #22 or the #37 plus a sweetener, I tell them that i'll look forward to having an opportunity to sign him after he's cut and walk away.
Add in that he only had one halfway-decent season and another that he spent half injured and the other half sucking, there's no way you drop a #4 overall on him.