But I am also interested to see if anyone had work around it.
I printed off a couple of copies and used them on two separate rooms (for the same nights) at the Crystal Gateway Marriott in DC. In fact, I actually did it twice. I had two rooms for the weekend we arrived (using the BOGO free EEOs for both rooms), then moved to the Crystal City Marriott during the week. The next weekend, we moved back to the Crystal Gateway Marriott and again used two more BOGO free EEOs. So I used a total of 4 printed out EEOs during that stay. (And if I remember correctly, they even upgraded one of the rooms to a suite room.) ^
But that was for spring break back in 2007, so things may have changed since then.