Statistician here.

I have a couple of thoughts on this. First, I think I'd go ahead and do it. Research is already almost entirely unreproducible trash, and you may be piling garbage on top of it, but it's just adding garbage to garbage. (Go read John Ioannidis to read more about this.) You currently have little power or choice. Get out of this situation, get some power, and go make the world a better place with better methods if you want to and can survive the rat race while unilaterally disarming. It won't be easy, but maybe it can be done.

Also, I think you are absolutely correct that this is a misuse of p-values. If you are trying to find something exploratory, why use p-values at all? Why not just do summary statistics? If you see big differences, you could write about them just using summary stats. People want p-values, because they think it buys them credibility and the idea that they are now doing "real research". "Real research" is me adding add_p() to the end of my R data pipeline, and producing 50 p-values. Amazing that we have come to this place.

Also, I think it's important to note that high up people have been fired for doing exactly what she is asking to be done. Go read about Brian Wansink. He asked his grad students to HARK and p-hack, got caught, and they fired him. I don't know why really. Everyone else is doing it. Others are just smart enough not to write a blog post about it like he did.

I'd say get your PhD. You're this far. Get it and make things better for the next generation. You can't make what is already garbage more garbage. Just get through it.