The other answers have gone quite sideways with this. Firstly let's cover the easier parts of this answer (and some of the comments) before we get into the technical stuff. So you're probably pulling this from "The Transporter" 3 or something similar movie. Why you wouldn't want to breathe from the car's interior is the petroleum products are all lighter than water. As the water displaces the air surrounding components -including the engine- it is going to strip these products off and extract them from the car making the air also contain more and more concentrated fumes. Since I rarely see people wash their engines as a mechanic, that is the reality.
However, the air in the tires isnt going to be much better. It has been sitting inside a petroleum product that experiences heat cycles. Anytime you let air out of a tire it carries that distinct smell.
Air is more buoyant than water. It is also compressible. All air-filled tires flex, no matter how rigid. If this wasn't the case, pressurizing them with air would be unnecessary to help give them strength to support a vehicle. So no matter the depth, as long as air pressurized before being submerged remained in the tire you could draw air out with your biggest problem in tripping the valve.
Now onto the bead popping. I like your idea, but you're terribly, terribly wrong. That device used to break the bead is done on tires that have had their pressure equalized and can allow additional air to escape. Also the hand lever models always include at least two rotational pivots, there by multiplying their force much higher than you think. Let's use the Audi A8D3 from the aforementioned movie. The math will be simplified to save me keystrokes. First you take the tire size 235/50zr18 and translate that into a rough cylinder. So you have a cylinder about 9.25 inches tall and a diameter of 22.25 inches. Then you punch out the rim to be left roughly with the tire. Figure the outside surface area of the tire, knock off 50 sq in for the change due to tires thickness and subtract the outside area of the rim's surface that the tire covers. You're left with roughly 1500 square inches. To seat a tire you need about 65-80 psi on average to get the job done. Since this is the closest that you could get to equal pressure exertion to unseat the bead -in theory- you're looking at roughly 105,000 pounds of force exerted at 65 psi. So that's about how much pressure you would expect to have to use to unseat a bead uniformly if it only contained ambient air at sea level right after it was seated. Remember, using said tires is going to introduce variables that will effect how much it resists. One part- no more tire lube. Yeah, that's a thing. Heat cycles will cement it to the rim too. Youre not looking at this being any easier to do once it hits the road. So with that external pressure requirement on an unpressurized tire, my educated guess is that it will have to descend much more substantially to make that happen. Also dont forget the temperature change as it travels through the water. As it cools, the tire will constrict even more- making it that much more resolute to keep its interference fit on that rim.
Should you do it? That's a personal call. A properly paranoid person would have a SCUBA tank stashed somewhere in the car. One that is well maintained and filled. If you aren't one of those, you still gotta breathe. Just make sure you've always got your knife and never forget your towel.