I agree that the "intuitive" answer is that the confidence interval will be "narrower" as the sample sizes increase, but how can we prove this more rigorously? My method above doesn't quite seem to work...

The trouble is that as n1 and n2 increase, the numerator ∑(Xi-Xbar)^2 + ∑(Yi-Ybar)^2 also increases (because now we are summing over more and more non-negative terms as n1 and n2, the upper indices of summation, increase), and this tends to make the CI wider. How can we deal with this matter?