I feel your pain having worked for companies with similar requirements, but do you really want to be the person that intentionally breaks travel policies so you can get airline/hotel points? I guaranty if you read your company's policy it will specifically mention it is against the rules to use a personal card so you can get travel rewards. It's not like nobody is going to notice that you are continually getting large reimbursements. Are a few measly travel points really worth the possibly consequences of flagrantly breaking company policy?

As with anything, if you really want to do this, why not just ask the company for it's okay? Of course you wouldn't ask, which already tells you what you want to do is wrong.