Having dealt with these issues a lot:
1) No, you cannot order someone to use their personal credit card.
2) Probably too late now, but the unit should have a DTEC (essentially a unit travel card that allows you to charge any unit travel expense, including rental cars). This is a separate card from the unit acquisition card, but the governing principles are similar. You could easily include the cost of additional drivers on this card.
One danger that a lot of folks don't realize is that should you use a personal credit card to rent a vehicle and end up in an accident, the insurance coverage that may be included with your personal credit card may not be valid if the credit card company realizes you were using it for DND business and NOT for personal use. I used to rent cars on my personal credit card (to get the points), but I don't do that anymore. I use my IDTC for that now (still rent hotel rooms on my personal card though).