How to verify my calculated amplitude under gauge invariance structure?

Gauge invariance requires that the amplitude vanishes if you replace the polarization with its corresponding momentum. If we replace $epsilon_4 to p_4$, we get begin{align} {cal A} &to 2 e^2 left[ frac{ p_3 cdot epsilon_1 p_2 cdot p_4 }{ p_2 cdot p_4 } - frac{ p_2 cdot epsilon_1 p_3 cdot p_4}{ p_2 cdot p_1 } + epsilon_1 cdot p_4 right] &= 2 e^2 ( p_3 + p_4 - p_2 ) cdot epsilon_1 &= 2 e^2 p_1 cdot epsilon_1 &= 0 end{align} In the second line, I have used the fact that $p_3 cdot p_4 = p_1 cdot p_2$ which is implied by momentum conservation. In the third, we have again used momentum conservation, $p_1 + p_2 = p_3 + p_4$.

You can try to verify gauge invariance by also replacing $epsilon_1 to p_1$.