On analogues of Weber's formula

Question

Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that $$ \int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta}). $$ One of the interesting features of this integral is the fact that it goes to zero very fast when $\alpha$ and $\beta$ are far away from each other. More precisely, we have $$ 0<\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx\leq \exp(-(\sqrt\alpha-\sqrt\beta)^2). $$ In some sense, this shows orthogonality of $J_0(2\sqrt{\alpha x})$ and $J_0(2\sqrt{\beta x})$. There are many proofs of Weber's formula. My favourite one (in context of this question) uses the Mellin transform. Namely, the Mellin transform of $J_0(2\sqrt{x})$ is equal to $\frac{\Gamma(s)}{\Gamma(1-s)}$. Taking the Mellin transform of the integral above over $\alpha$ and $\beta$, we get $$ \int_0^{+\infty}e^{-x}x^{-s-t}\frac{\Gamma(s)\Gamma(t)}{\Gamma(1-s)\Gamma(1-t)}dx=\Gamma(1-s-t)\frac{\Gamma(s)\Gamma(t)}{\Gamma(1-s)\Gamma(1-t)}. $$ Using Taylor expansion of $I_0(2\sqrt{\alpha\beta})$, one can reduce the proof of Weber's formula to the relation $$ _2F_1(s,t,1;1)=\frac{\Gamma(1-s-t)}{\Gamma(1-s)\Gamma(1-t)} $$ for the Gauss hypergeometric function. My question is, can we generalize this to other similar functions? To narrow this down, let $$ g(x)=\frac{4}{\pi}K_0(4\sqrt{x})-2Y_0(4\sqrt{x}), $$ where $Y_0$ and $K_0$ are ordinary and modified Bessel functions of second kind. Then $$ \int_0^{+\infty}g(x)x^{s-1}dx=\frac{\Gamma(s/2)^2}{\Gamma((1-s)/2)^2}. $$ Can we choose some nice (smooth and positive) function $h(x)$ so that the integral $$ H(\alpha,\beta)=\int_0^{+\infty}h(x)g(\alpha x)g(\beta x)dx $$ is very small when $\alpha$ and $\beta$ are far apart? (here I want something very similar to the example above: optimistically, the same $\exp(-(\sqrt\alpha-\sqrt\beta)^2)$ estimate) Furthermore, can this be deduced from some hypergeometric identity?