Can't believe nobody said this yet, but welcome to Head Fi and sorry about your wallet!
The K/Q701 series always sounds a bit harsh & plasticky out of the box. It definitely does improve & smooth out over the first hundred hours or so. I also like to put mine in a closet just burn it in for a few days before I hear a new pair. But it will improve over time regardless, and you'll probably end up getting used to the bass because it is clear and precise, rather than bloated. It's just kind of a habit thing, like enjoying too much salt on your food than you really should.
Schiit always recommends the Lyr for this series of headphones, period.
I have always used a tube amp with mine. I think the headphones sounds terrible coming out of the solid state amp, no matter how many I tried. Its just too harsh and analytical, while a tube amp softens & warms it up and makes it beautiful and musical again. So that's my advice, definitely shoot for a tube amp in the future.
I agree with the DSP idea if you really need it, but I think over time you will really appreciate the headphone's sound signature just how it is.
The Q/K and the DT 880 are variations of a theme. Also, the DT comes in 3 models which are all quite different. The 600-Ohm is by far my favorite. If you really want to know for sure, you really need to listen to them side by side. Both headphones have their weaknesses and strengths, and in the end I chose the AKG because of its much better soundstage and instrument separation, and let go of the DT which had more impactful bass. The AKG goes deeper and is more controlled, but the DT had much more force. It was something I just learned to live with.
Have you even tried the Q701? The Q701 and K702 are not the same and shouldn't really be clumped together. The Q701 has never been "harsh and plasticky" EVER for me. Sure, if the recording is harsh/bright then the Q701 will reproduce this, but it will never be fatiguing.
It makes no sense why a Q701 would need a tube amp to sound good. They're perfectly fine for most with a solid state amp. They don't need to be warmed up at all IMO. The Q701 has never been analytical/cold/thin to my ears. That's more like the DT-880 to me.
On my Headroom Micro (very transparent and more so than the O2 IMO) the Q701 is very smooth and full sounding. This depends on recordings. Some of them will sound bad no matter what. There's some tracks I have where it feels like all the bass and warmth is sucked out of them. Not the amp, DAC or headphones fault.
BTW my Q701 sounded great out of the box. There is a chance it might have improved later on, but no way to know for sure.