If all the beneficiaries are of full age and capacity, they can reach whatever decision they wish on doing a DoV to add the exec as a beneficiary. There is nothing to stop the two who wish to include him varying THEIR shares so that the two shares are split three ways (ie so he takes a share from their inheritance only). However, if the third objects, he cannot "contest" the decision. He can only contest the will on the basis that for some reason it is invalid. If he has "proved" the Will (ie obtained probate), this will prove extremely difficult for him since he will have sworn an oath to say it was the last valid will of the deceased.
In terms of compensation from the hospital, he as the executor has the duty to pursue this claim. However, if he refuses, it is possible to either have him replaced as an executor or for the beneficiaries to run the claim themselves. As an executor he is in a position of trust to act for the benefit of the estate and not for himself and it is settled law that unless the will provides otherwise, an executor should not benefit personally from his office. However, this latter point depends on to whom the compensation is paid. Depending on the type of case it is, it could be paid to the family of the deceased or to the estate (for instance, a claim for the loss of a parent in difficult circumstances could mean he has a personal claim on the other hand, a claim that the deceased suffered would be paid to the estate).