Could they plug the hole with a pig. They use these things every day for purging and testing pipelines. They can't seem to overcome the pressure with mud, so a BOP could be placed on the current BOP and a pig sent in with air pressure behind it. They should be able to generate more than 5000 PSI. If it works they could start the cement right away. The biggest challenge is getting the pig in.
1) If they did place a BoP on the BoP, don't need the pig, then they could pump the mud right down and would not have alternate escape path like it has now.
2) have to generate more like 15,000 PSI (and then tranfer it down to the new BOP). Even with a pig, mud woudl be better pushing force
3) Getting the pig in would be the tricky part 9as you note) and more difficult than injecting mud via new BOP. (Also most of the work on the new BOP could be done before it was dropped to the bottom. Much faster).
4) You actually WANT mud under the cement. That mud helps hold teh cement in place even after the cement cures.