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I already have one. Thats what enables me to identify the exact frequencies I find harsh. But my own experiments show that EQ cannot compensate a bad crossover. This implies that the problem isn't really frequency response but phase response.
I think it's more likely to be poor frequency response integration (particularly off-axis, even if on-axis the speakers measure flat), since there is no published evidence of anyone having been able to hear the degree of phase distortion likely to be introduced by the speakers you've been auditioning.
You can test yourself on this, however, with this neat little ABX tester called.
The UI is not great, but what it basically allows you to do is to load any audio file of your choosing and then ABX test yourself between the original unaltered file and the same file with the a given degree of phase distortion similar to that introduced by a crossover at a given frequency.
I'd suggest you try this to rule out phase distortion as the culprit. It's a bit fiddly, and making sure you have extremely low latency is important to ensure you don't get false positives. Best to use it with single-driver headphones.
If indeed phase distortion is the issue, then I'd suggest looking into some linear-phase active speakers like those from Kii or Dutch and Dutch, if they're not too far outside your budget.