Why does my attempt at RSA break?
Since your q is composite with factors 3,7,19 we get $\phi(q)=(3-1)*(7-1)*(19-1)=216$ and not 398 The rest of the calculation follow as in the question. 

And since $p=2$ we get also $\phi(n) = 216$

As a result we also get $d=e^{-1}=7^{-1}=31 \bmod 216$

And we can verify that $669^{31} \bmod 798 =123$ as expected.