I believe that you can get rid of the immobliser, one of the options being replacing it with a replacement alarm/immobliser system, there are ones that are designed to get rid of the Bosch 951 immobliser/alarm system.
However, like Rich said, it would probably be cheaper to get hold of a second hand unit (with several remotes, as ford charge 56.00 per additional remote and an hours labour to program more) and see if that fixes this problem.
The only way to program more remotes is through ford, none of the procedures that the RAC/AA have on their computers for the Bosch 951 system work with the Explorer, as it will only respond when you enter the unit code.