Been there, done that, have the t-shirt to prove it. The days of $500,000 annual budgets are gone. Group owners are becoming more and more cheap almost daily and this is the crap that you must either come to accept or look outside of this industry.
I've done the same type of promotion, "The Ten Thousand Dollar Secret Sound." The problem is that the listener needed to specify the particular sound, i.e: uncooked rice falling on aluminum foil. I'm not kidding. The prize... $1,000! There were ten total prizes budgeted for $1,000 each. I believed this contest to be deception. Lost the argument to an out of town consultant and a Johnny-come-lately GM that only remained at the cluster for a total of three months.
Sadly, this has become the state of the industry.
Make sure your rules are bullet-proof, clearly spelling out the details of the giveaway including the premise of the contest, the number of prizes, and the methodology for prize giveaways. Use your website to help clear up the terms of the contest by offering additional details that the on-air promo doesn't. Provide the official contest rules at the studios AND on the website (if you don't do this already.) The on-air promo may be ambiguous but remember to dot the I's and cross the T's in your contest rules and you will be fine. The contest rules are your legal framework of your promotion and provided you stay outside of the "lottery" parameters of the FCC, what you as the station has to say goes.
I understand the ethical quandary that you're facing. I've done broadcast promotions for fifteen of the best years of my life. Eventually, the smarmy requests from unethical consultants and poorly qualified General Managers led me to look elsewhere for challenging careers. There is life outside of radio. Hang in there.
