I wonder if comparing powers of sums would help?
Suppose one of the sequences was 3, 7, 15, 30, and consider s = √[3] + √7 + √15 + √30.
Then s^2 = 55 + 30 √2 + 6 √5 + 6 √10 + 2 √21 + 2 √105 + 2 √210.
s^3 = 159 √3 + 90 √6 + 151 √7 + 90 √14 + 135 √15 + 105 √30 + 18 √35 + 18 √70.
...
s^30 = 1679613741139712617544067791402275 + 1187666266098277047375460186425450 √2 + 750901847525954802822187362608010 √5 + 530967788407084153582901906991810 √10 + 366402251460399628674629181584238 √21 + 259085516641872377051783075481000 √42 + 163913368573924563188162165414670 √105 + 115904254449237925942667189567670 √210
and so on.
For any given finite sequence of positive integers there is a finite set of square roots of positive integers such that all positive integer powers of the sum of the square roots of the sequence members can be expressed as a linear combination of that finite square root set with non-negative integer coefficients.
With s^n if we approximate each of the square roots by first the nearest integer not larger than the square root, and second by the nearest integer not smaller than the square root, we get a lower bound and upper bound for s^n.
Suppose we do that for the sums of the square roots of the two sequences we want to compare. For each power n, that gives us two integer ranges we can compare. If they do not overlap, we can tell which square root sum is lower.
If they do overlap, we can try a higher power n, so we can try a better square root approximation than nearest integer to get narrow ranges for the n'th powers.
What I'm hoping for is that by going to higher powers we can avoid having to do high precision square root calculations, so it can be done with just large integer arithmetic and regular floating point.