A bit long for a comment, just checking the formulation of the problem. For $n=6$ I get $$ 4 a^3 b^2 = 3 a^2 + b^2. $$ Then for $n=8$ I get $$ 2 a^3 b^3 = a^2 + b^2 .$$ These are likely to be the two easiest. Anyway, there are sometimes methods for ruling out the existence of rational points on these curves other than the automatic $(a=1, \; b=1).$ But this really does appear to be a different problem for each $n,$ meaning your question is a large project.

A bit long for a comment, just checking the formulation of the problem. For $n=6$ I get $$ 4 a^3 b^2 = 3 a^2 + b^2. $$ Then for $n=8$ I get $$ 2 a^3 b^3 = a^2 + b^2 .$$ These are likely to be the two easiest. Anyway, there are sometimes methods for ruling out the existence of rational points on these curves other than the automatic $(a=1, \; b=1)$ and the rather artificial $(a=0, \; b=0).$ But this really does appear to be a different problem for each $n,$ meaning your question is a large project.