I managed this with a hint from a friend at a key point.

Assume there's a solution.

Let a1-a6 = probability of rolling 1-6 using die 1 and b1-b6 = probability of rolling 1-6 using die 2.

P(2) = a1*b1 = 1/11.

P(12) = a6*b6 = 1/11.

(Hint from friend follows. Thanks, friend.)

P(7) = a1b6 + a6b1 + some positive terms = 1/11.

So a1b6 + a6b1 < 1/11

a1(1/11a6) + a6(1/11a1) < 1/11

a1/a6 + a6/a1 < 1.

This is not possible with positive values, so the assumption there is a solution is false.