If you have any material amount of play at all at the CET properties in AC, it doesn't take too much to get a comp room offer on off-peak dates at their lower-end properties in Vegas. Whether you want to stay at those properties is another matter. And if you take any of these comps or benefits and don't give the property a proportionate amount of play during that trip, you'll be penalized on future offers, though it sounds like you aren't really trying to play the long game here.
If you do plan to try to build any degree of comps / loyalty with any of the casino chains, it could be worth switching hotels between your work travel nights and your extension nights. Vegas hotels tend to look at your customer value over the course of your stay, and measure you based on amount gambled per night stayed (roughly). So if you have a few nights there for work and don't gamble those days, but they are part of the same reservation, any gambling you do on your added day would be "diluted." Though again, if you're not really expecting to build up any long-term loyalty with any of these chains, it may not matter too much.