If you are required to pay for your room with the corporate card, you must pay for the room with that card. Subbing the card at checkout is expressly what you should not be doing.

On the other hand, if you are not required to use the card for meals at the hotel, do not charge them to the room. Simply use your Marriott card in the hotel restaurant and those will never appear on your room bill. Then you won't have to go through the time-consuming mess of having a separate folio created.

As to other schemes, you first need to speak with your travel people to find out the reason for the requirement and then make a decision. Some companies are dead serious about this stuff and terminate people for violations. Others say they require it and could care less and haven't enforced it for 20 years.

As others note, this is about corporate culture and you need to figure that out before plotting next steps. It would be a pity to find out that you can break the rule and nobody cares and equally a pity to find out that if you break it, you lose your next promotion or some nice bonus.