I'm sorry for the delay, but I have a lot of travel in August.
I don't know whether it will work, but I hoped you might be able to scale the integral as follows.
Let h be the integrand. Then \integral h(x) dx = h(0) * \integral (h(x)/h(0)) dx.
Let g(x) = h(x) / h(0). My hope was that the integral of g is better behaved an remains bounded as your parameter varies. The value of the integral should be small for all values of the parameter. However, I haven't had time to think about this problem recently or to see whether my idea actually works.
A double integral of a convolution is just too hard for a casual observer to offer much specific help. It would take me quite a bit of effort to understand the details of your problem. I still think your best bet is still to simplify the problem and understand each component. The key is understanding why the integral isn't converging. For example, I notice that the documentation example for the QUAD function says that the inner integral should not range over an infinite domain. Have you tried integrating the inner intergral on a finiate domain such as [-10, 10]?
