My employer has some exemptions from the corporate card requirement intended for low-expense infrequent travelers (something like 2 or less trips and less than $1000 spent in the year, IIRC). That may be something to look into--unless of course you travel frequently.

Another possibility if you use a corporate AmEx is to book hotels that don't accept it. While this may be difficult/impossible to do at Marriotts (and you may be unable to do so at all if there are other rules heavily encouraging/requiring stays at specific locations), corporate travel policies do tend to have exceptions in the event the card isn't accepted at an establishment. Double check with your own company's written policies before attempting, of course.