Biden's health would be a significant problem if he continues his presidential campaign. In OTL he suffered two brain aneurysms in 1988 and had two surgeries, one in February and one in May, to fix each of them. His campaign had ended several months before then, in September 1987, but even when he was campaigning the symptoms of the aneurysms were apparent, with Biden experiencing frequent headaches and neck pain. The first aneurysm nearly killed him; a priest actually administered Last Rites to him on the hospital bed. Biden was away from the Senate for seven months in 1988 while he was hospitalized and undergoing recovery from the aneurysms. His wife and sister believe that if he had continued his campaign, then he might not have been able to receive the medical attention he needed in time. He would have literally campaigned himself to death.
This, in my opinion, is the biggest obstacle to a continued Biden '88 campaign. Even if he seeks treatment for his aneurysms before they get worse, he will be out of service for many months to recover. He won't be able to campaign effectively if he's confined to a hospital bed or going through rehabilitation.