My approach is slightly different.
I don't want to antagonize anyone, nor do I take the discount request personally.
However, it is my responsibility to pivot away from their poking at my pricing.
So what I usually do, is challenge their proposal: "I don't see how I could deliver my usual quality on that kind of budget. In order to keep the price under $x, we would need to cut quite a few corners and to be honest, I wouldn't want to compromise on your vision."
Then I suggest alternatives that don't involve me, like another fiverr freelancer or what to type in the search bar to get something that matches their budget.