Times have moved on. In 1936, the Church of England was still totally against divorce. It would have been unthinkable to have a King married to a divorced Queen when the British monarch is also always Supreme Governor of the Church of England. But the real reason behind it is that Edward VIII was not really interested in the official paperwork side of the job, tended to leave secret papers lying around, and was pro-Nazi. The "Establishment" wanted him out, and the position of the Church of England on divorce was a convenient excuse.
The Church of England now permits its priests to conduct remarriage of divorced people, though it's still not entirely accepting of it. Being married to a divorced person is no longer the barrier it used to be. This ambiguous position is why Charles and Camilla married in a register office and not a church, though the Archbishop of Canterbury conducted a service of blessing afterwards.
There is only one way in English law that Charles could stop himself becoming King, and that would be if he converted to the catholic church.