I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception of one (which I've linked to in this post).

The integral I would like to investigate is the following:

$$\displaystyle \int_{\mathbb{R}^2} \frac{J_{1}(\rho \|x\|)J_{1}(\rho \|b-x\|)}{\|x\| \|b-x\|} \ \mathrm{d}x,$$

where $J_{\nu}$ denotes the Bessel function of the first kind, $b \in \Gamma \backslash \{0\}$ for any rational lattice $\Gamma \subset \mathbb{R}^2$ of full rank, $\rho > 0$ is large and independent of $b$ and $x$, and $\|\cdot\|$ denotes the standard Euclidean norm. I'll try to explain from where this integral arises, and why I care about it.

This integral arises when considering a problem on the distribution of lattice points inside a ball in $\mathbb{R}^d$ with radius $\rho$, which can be thought of as a generalisation of some aspects of the Gauss circle problem. For the full details, I recommend this paper (particularly pages 10-11 and 15-16). The paper considers $\sigma_p$ and proves some asymptotic bounds for $p = 1$ and $p = 2$ which depend on the dimension $d$ and the radius $\rho$. I'm attempting to generalise that work to $p = 4$, and with some extra work, any $p \in \mathbb{N}.$ The derivation of this integral begins here, and uses the estimate at the bottom of this post to arrive at the aforementioned integral. With the help of this answer, I was able to prove that the integral

$$\displaystyle \int_{\mathbb{R^d}} \|x\|^{-d/2}\|b-x\|^{-d/2}J_{d/2}(\rho \|x\|)J_{d/2}(\rho \|b-x\|) \ \mathrm{d}x,$$

converges absolutely in all dimensions $d \geqslant 3,$ after using the asymptotic bound for the Bessel function, $|J_{\nu}(z)| \leqslant C|z|^{-1/2},$ for $z \rightarrow \infty.$ However, as the answerer explains, the proof he provides does not work for the case $d = 2$. The integral at the top of this post is therefore the remaining case left to verify. I would like to know if it converges, and how in particular it can be estimated.

I don't have much experience in dealing with integrals involving Bessel functions. I would appreciate any help anyone can offer on how to deal with this integral.