I must not be understanding the problem here because this seems pretty simple. If someone told me to add two lists of numbers (the square roots) and then take the difference of the two sums and the sums were potentially too big for the computer to handle, I'd start differencing before I finished the sums. (For example find the total of sum1 - sum2. While processing values, if the running sum is positive take a number from list 2. It it is negative take a number from list 1.)

That should be linear in the total number of elements in the list.