Copyright infringement is not suitable as an analogous case because the law specifies statutory damages for it, so proving damages is not typically necessary for the types of works which you are thinking of.

Let me give a detailed analysis with some concrete, but arbitrarily-chosen numbers, and then I'll show you what a lawyer representing Amazon would say to attack the argument you've presented.

Suppose you notice that 5 per cent of people whom you ask to subscribe to your mailing list actually subscribe (it is almost certain a real number would be much lower). Then, of those who subscribe to your mailing list, 10 per cent of them make a purchase when you send an advertisement to them through that mailing list. And then, of those who make a purchase, the average sale is $50, of which $20 is profit. Therefore, you argue damages of 5% × 10% × $20 = $0.10 per customer. Suppose Amazon placed 1,000 orders this way. You therefore plead damages of $100 (the fact that this is a trivial amount is not relevant to the legal analysis).

The legal method for the calculation of damages is to compare what your financial situation would have been had Amazon not done the thing they were not supposed to. Amazon will argue that had they complied with your terms of service, 0 orders would have been placed as you forbade AI agents from placing orders, and therefore the profit can be calculated as 5% × 10% × $20 × 0 = $0. After this argument is made, it then becomes your burden as the claimant to rebut it. You will have to prove what percentage of people ordered through Amazon, who would have otherwise ordered from you directly (and thus you would have the opportunity to advertise to). This is a fundamentally very difficult task. Amazon would probably propose to the court that you ask all of the customers to testify that they would have otherwise ordered from you directly, and then you can count it as ten cents per witness.

All of that notwithstanding, Amazon will still argue your damages are zero, because you have not actually lost the ability to connect with the customers they have given you, because you still have the ability to ask them to subscribe to your mailing list by including a card to that effect in the package you send them. The fact that both of us very well know that nobody will do that is not legally relevant: the action is possible and the law does not particularly care about whether it is easy or effective.

I know it's tempting to call me a bootlicker or whatever, but the fact of the matter really is that the law is not favourable to the claimant in this case. This is just a bad argument to make with no sufficient legal justification to claim anything more than a nominal amount of damages. Yes, Amazon are a bunch of assholes, but sometimes, being an asshole really is legal. The law is not a proxy for morality and the courts are not infallible guardians of justice. They are institutions that interpret fallible, imperfect, human-made rules.