There is no law against not using a statistical test at all when the effect you see is not meaningful to you. The purpose of statistical tests is to see if a result that could be important is real, or due to chance. That isn't the character of what you need to do here.
I'll add that if it's really important for you to do a test, you could also test for equivalence: set up a threshold value, $\delta$ whereby any difference less than $\delta$ is deemed unimportant. Then do two one-sided $z$ tests -- one of $H_0^1: \mu_1-\mu_2\ge-\delta$ and the other of $H_0^2: \mu_1-\mu_2\le\delta$. If you reject both hypotheses, you'll have strong evidence that $-\delta<\mu_1-\mu_2\ <\delta$. The easy way to do this is to calculate a $(1-2\alpha)$ confidence interval for $\mu_1-\mu_2$, and reject both hypotheses if the interval lies entirely within $(-\delta,\delta)$.
It is possible - especially with large datasets - to have both a significant difference and significant equivalence, so you need to be prepared to explain why that can happen.
Addendum: Testing for "supersignificance"
Yet another idea -- and this may be closer to answering the original question -- is to turn that equivalence the other way, and test
$$ H_0^\delta: |\mu_1-\mu_2|\le\delta \qquad \mbox{vs.} \qquad H_1^\delta: |\mu_1-\mu_2|>\delta $$
This is saying that a difference of $\delta$ or larger is required to be considered important. This can be tested via the test statistic:
$$ z(\delta) = \frac{|\bar y_1 - \bar y_2| - \delta}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} $$
The decision rule is to reject $H_0^\delta$ in favor of $H_1^\delta$ if $z(\delta) > z_{\alpha/2}$ (e.g., $z(\delta)>1.96$ if using $\alpha=.05$). Note that if we choose $\delta=0$, we have $z(0)=|z|$, the absolute value of the usual $z$ statistic, and this is then just the usual $z$ test.
Interestingly, the same statistic can be used to test for equivalence, for which the null hypothesis is $H_1^\delta$ and the alternative is $H_0^\delta$ (!!!). Specifically, we reject $H_1^\delta$ in favor of $H_0^\delta$ if $z(\delta) < -z_\alpha$ (e.g., $z(\delta)<-1.645$ with $\alpha=.05$). If $z(\delta)$ is between these two critical values, we have an inconclusive result, and clearly it is not possible for both to be significant. Note also that $z(0) \not< 0$, so you can't conclude equivalence with a threshold of $\delta=0$.
