TBH .. If you really want to squeeze out more quality, then I would look to Audeze LCD-2 or LCD-X.
Best by getting the Alcantara version, because this has positive impact on the final result.

The SRH 840 is already a quite nice phone for mixing and mastering, see recording.de review which has been done by a mastering engineer. According to this review the SRH840 is not that bad for this purpose.
https://recording.de/magazin/vergleichs … oert.2621/

So the basic question is what you dislike in regards to the Shure ???
And throwing in only around €70 Euros more really won't make any big difference it's then more a matter
of taste and this you can only find out on your own by comparing the phones.
For this you need to be able to quickly change between them back and forth and taking excellent audio content
which you know very well.

Other phones in that consumer price range do not deliver adaequate to the ADI-2 DAC capabilities.
And I think the next step towards higher quality is most likely to look in the area of planar phones.
But also this is a wide range and not everything is perfect only because it costs a lot more.

I tested different phones approx 2 years ago on an ADI-2 Pro: https://www.tonstudio-forum.de/blog/ind … ADI-2-Pro/
Here the complete PDF of my review: https://www.tonstudio-forum.de/index.ph … -v1-0-pdf/

If you can effort, check out an LCD-3 but really the Alcantara version (which is sadly not available anymore, but maybe you can ask whether you can still get one) and then take the more comfortably headband from the LCD-4 on top.
Alternatively get the LCD-X which you can get to a very decent price at the moment.
But I do not like the sound representation compared to the LCD-3 and it doesn't show the details of the audio material like the LCD-3 does.

What do you think about HiFiMAN HE-400i https://www.techhouse.sk/produkt/hifiman-he-400i/ ?