Infinite $\beta \eta$-reduction sequence implies infinite $\beta$-reduction sequence
As usual, the bible knows all, and Lemma 3.3.8 (p 66) gives the detailed proof of commutativity of $\beta$ and $\eta$ reductions.

Trying to reproduce it here would be tedious and wouldn't do it justice, so I'll include a picture:



You had the main idea right, it just takes a bit of practice to generalize from the specific examples.