If you’re not making this up, this will backfire on you massively. If he has already got legal advice, and like another poster that mentioned it earlier I strongly suspect he has, then he will also be looking to establish 50/50 by way of a court order. He’ll get that, too.
Do not think that your ‘strategy’ hasn’t been seen thousands of times before. It has, and it’s transparent. You will, correctly, be viewed as acting in your own interests rather than those of your children who have been cared for well prior to this, with their parents having an amicable co parenting relationship. That you have now decided to go a hostile route and blow this up because of your own greed, and again they’ll know that’s exactly what this is, does and will not reflect well on you. Lying especially won’t. You’re wasting your own, and most importantly their, time.
He already has your written word, evidence of own expenses relating to your shared children that will reflect the amount of time in which they are in his care (clubs, and childcare for example), and likely other correspondence that can and will be used to back up his claim. Also, considering that he already finances private medical and dental care on top of whatever else he pays for during his time with them, I would not be surprised if his expenses exceed yours proportionally too, so don’t think “I bear the greater burden” is even necessarily to be an argument that you can use, if they were even inclined to entertain it at all (unlikely). This is simply procedural, and not something they’re going to be at all interested in wasting their time with.
By all means go ahead, as I’m sure you will, no one hear actually has any vested interest in stopping you from shooting yourself in the foot. Tbh I think you should go ahead, a hard head makes for a soft ass and it’s going to be a useful lesson.