Actually the CoE could marry them, if the presiding minister so chose (mine would have done so with no problems at all). The issue wasn't about Andrew still being alive but how much Camilla contributed to the breakup of the first marriage of Charles and Diana.
I believe the issue rather concerned the Prince of Wales as the future King, and had very little to do with Camilla's role in the breakup of his first marriage. Although Charles and Diana divorced, that was no longer an issue at all. Diana's death effectively made Charles a widower in the eyes of the Church. What did matter, though, was the fact that Camilla was divorced and Andrew Parker-Bowles is still very much alive.
The Church of England revised their position on re-marriage in the church after divorce in 2002, but it is important to realize that this change does not confer an automatic right on couples to further marriage in church.
Above all, the system now in place is designed to safeguard the Church's doctrine of marriage. While the decision rests with the officiating cleric, he/she has the option of referring cases to the bishop. The Church of England's view on marriage has not changed.
They do recognize that in the 21st century, divorce is a common practice, and they will try to accommodate their parishoners within the framework of their doctrine. They admit that there are exceptional circumstances in which a divorced person may be married in church during the lifetime of a former spouse.
In the case of a Prince of Wales, however, the situation is entirely different. He will one day be the Supreme Governor of the Church of England, and it is crucial to that position that he uphold the doctrines of the Church.
I believe that not allowing him to marry Camilla in church because of her status, was a decision made to protect his future role as Supreme Governor.
If he were an ordinary parishoner, sure.. the minister may have married them in church.. but the Prince of Wales in no ordinary parishoner.