Barr may have resigned because he declined to appoint a special counsel and instead wanted his successor to take the fall for this act of revenge (for Biden's role in the attempted Flynn extortion scheme)... which seems very likely. Of course, that special counsel would have as much leeway to investigate Hunter as did Mueller's henchmen when they were supposed to be investigating the Russian collusion kerfuffle... ie., almost total! So what can Biden do in response?
Well, the rules drafted in the wake of the Starr investigation of Clinton say that a special counsel “may be disciplined or removed from office only by the personal action of the Attorney General for good cause." In general, federal agencies can only remove the employee for misconduct, poor performance, or medical inability to perform. Additionally, prior to effecting the removal, an agency must provide the employee with due process including notice of the proposed adverse action, production of the evidence relied upon, the right to submit written and/or oral replies, and the right to be represented by counsel. Further, an agency must be able to show that an employee committed either the misconduct as charged by preponderance of the evidence (i.e., more likely to be true than not) or had poor performance case through substantial evidence (i.e., less than 51%).
The bigger question is whether our media would cover the investigation and publish leaks like it did in the Mueller matter?