Actually, you can. The Amex Centurion (not Platinum; may have been a typo on jfe's part) gets you mid-tier (50,000-mile equivalent) status on Continental and one other airline I forget at the moment. Whether you'd qualify for one, since it generally requires a history of charging at least $150K/year on another Amex card, and whether it's worth the $2,000+ annual fee, are separate questions.

Beyond that, I'd try to make a case with your new employer that (a) status makes life on the road more tolerable for the reasons you mentioned and (b) it doesn't cost much, especially since they presumably can use up all their Amex points without trouble and it they save no more money using them on you than on someone else. (It would be different if total company travel couldn't use them all up, but I gather that's not the case.)

It's not like you're asking for paid F or anything. It's just that, for an occasional traveler, flying on an Amex reward or paid ticket makes no difference, whereas for a regular traveler it does, so why not use them where it does no harm and use paid tickets where it will do some good?