Save As PNG has a curious bug: It doesn't embed the sRGB profile. All other profiles embed correctly, but not sRGB.
To make matters worse: instead of showing up as <untagged>, which it should be because it is, instead it takes on your working space and assigns that.
In short, Save As PNG is an utter mess, and I don't go near it. Luckily, Save For Web and Export both do it correctly, so always use one of those when working with PNGs.
But there's one more problem. Export doesn't record in actions. So you can't use that. That leaves Save For Web. Record an action using Save For Web, and make sure to check both "convert to sRGB" and "embed color profile". The latter is unchecked by default. Batch that action.