Pretending pure math is applied math for grant funding

This won't be a direct answer to your query, but I hope will be useful advice. First, if you must do pure math and you must get funding to do it, then you have no option but to apply. If you don't apply, you don't get funded.

Next, being devious isn't going to help you and will likely be recognized, as you fear.

However, on some time scale, all mathematics is applied. The time scale may be long, of course. In my personal case, it took about thirty years for someone to find an application for my "beautiful" but extremely arcane dissertation to find application. I'd predicted that it would never be done. But thirty years isn't very long, so that doesn't help you directly.

You work in some subfield of mathematics, of course. It might be very narrow and it might be a currently popular subfield, or not. But it is likely that others in the same subfield, or a closely related one, have seen applications of their work to some problems. If you can learn what those applications have been, you have the basis for a statement that "Advanced work in X has been known to contribute to Y as evidenced by (citation)". That isn't devious, or phony.

But note that "applied" is actually a continuum, not a discrete thing. Pure is pretty far to one end of the continuum, but it isn't disconnected from the other. But it will take some investigation on your part, and I suspect that you are good at that sort of thing.

I'll give you my perspective as an applied mathematician who has read a fair share of pure math proposals. I do, for example, find it amusing that essentially every number theory proposal stresses the importance of the work to cryptography. One can conjecture that it's rather unlikely that that is actually true for even a significant fraction of these proposals, but it is the common approach in that community to address NSF's requirement to explain the "Broader impact" of the work.

Now, since that is so and everyone does it, it's likely that a selection panel consisting of people well versed in the field is just going to read over that statement and ignore it -- everyone knows that applicants need to pay lip service to the broader impacts, registers that they do, and moves on. In other words, as long as you stay within the norm of your subfield in conjuring up possible connections to applications, then there is no problem for you. Of course, if you promise completely outrageous connections, then even a panel of people who are in the same situation as you will find that you've gone beyond. Where exactly that line is is of course hard to define -- talk to others in the community and let them comment on what you write.

I do want to add one point to this, because I really don't want to come over as a pure-math bashing applied mathematician: The same of course happens in what is generally understood as "applied mathematics". Whether you try to show that the solution of some PDE is in a slightly smaller Sobolov space than was previously known, or whether you show W^{1,\infty} error estimates for finite element discretizations of some obscure equation, the truth is that much of this kind of research is also pure. The only difference is that the equation you are considering might have been motivated by some actual application, but the actual work you do is not motivated by trying to actually solve the problem in the same way as an engineer would want to approach this. In other words, your work may be closer to applications and is often easier to explain to laypeople, but it's not exactly applied in many cases and the "broader" impacts are also quite limited.