RG for Interacting Fermions - Shankar's Review

I want to ask a question about a rather famous review paper on RG by Shankar.

On page 92-93 of this paper, Shankar provides a version of RG for spinless fermions in $d=2$ dimensions with a soft cut-off (at $T=0$). This soft cut-off has completely baffled me. On one hand, it conforms to the intuitive expectation of allowed scatterings in the low-energy theory; on the other, its math leaves me fairly confused.

I include two images below from the arxiv version so that readers don’t need to go to arxiv, at least not for the purpose of answering this question. (In particular, if I am in violation of any rule, please feel free to inform me or edit it)

What I do not precisely understand is the ad hoc way in which this cut-off has been introduced. It seems this extra exponential suppression is added on top of the physical interaction. Then, as Shankar does the tree level RG, the physical $u$ remains the same (and only needs rescaling); while the exponential cut-off becomes even sharper, thereby effectively leading to exponential suppression of $u$. At least this is what Shankar’s math suggests. Is this understanding correct?

If the above is correct, then isn’t it paradoxical that Shankar’s tree-level exponential suppression comes from a factor in the measure that depends on the cut-off itself? I’d have expected that at tree-level the strength of such a particular interaction would stay intact, instead of being continuously renormalized because of extra factors that actually depend on the volume of integration itself.