Just by hand considering these triples (1000,1000,0) and permutations (2*pi, 4, 4) and perms (1,1,1) I observe that g1<g2+g3 is obeyed but g1<0.9999*(g2+g3) can fail;; and g1>0*(g2+g3) is obeyed but g1>0.0001*(g2+g3) can fail.
So I believe an inequality of the form g1<K*(g2+g3) is obeyed for some constant K>=1 and I do not know (but computer could pretty much tell us) what the minimum possible K is. The obvious guess is K=1.
--this crude MAPLE program g := (a,b,c) -> (max(a,b)*EllipticE(1-min(a,b)^2/max(a,b)^2), max(a,c)*EllipticE(1-min(a,c)^2/max(a,c)^2), max(b,c)*EllipticE(1-min(b,c)^2/max(b,c)^2)); K := 0.0; do for i from 0 to 10000 do z := evalf(g(rand(), rand(), rand())): K := max(K, z[1] / (z[2]+z[3])): od; print(K); od; output 0.9793366600 0.9992199627 0.9992199627 0.9992199627 0.9992199627 making it virtually certain that my conjecture is correct, i.e. g1 < g2+g3 the usual triangle inequality, also is valid for ellipsoid girths.