Again, as a dealer, I know this is a false bargaining chip. If it is such a HUGE deal, the customer would have walked away. Remember, we sell as many cars in a week that you buy/sell in 5 years. We know all the excuses, we can smell fake objections from a mile away. It's literally our job to do that. Allow me to present an alternative tactic.
OP, I didn't realize your allergy was a concern. I thought your concern was "was this a car that was abused with a dog constantly in it?" In this case, coupled with your mention that the car was noted to be unclean inside and out by your inspector, I would ask either that the dealer perform a deep deep clean, or that they discount the amount it'd take for you to get it over to somewhere like Esoteric to have it done to your satisfaction. Don't feign dismay. Nothing makes a dealer hesitate to discount a car like "well I really don't want this car BUUUUUUT..." What we'd rather hear is "well I like this car, but this is what it takes to earn my business. If you do this, then I will buy the car." It shows genuine interest. It shows that you're committed to holding up your end of the bargain. It gives them a clear path to the sale. It, most importantly, makes them confident that if THEY hold up their end of the bargain that you won't ask for more things.
I say this because I've let deals unwind after we performed a couple services that were conditional to the sale, only for the customer to then say "well I appreciate that you did those things, but I'm still not satisfied with the car and I'd like another $XXXX off".