If you want to go the Cavalli route i'd opt for their CTH amp instead. HD6XX is already warm sounding, I wouldn't want to pair that with a warm sounding dac/amp, unless that is your preference.

Yup, the massdrop Liquid Carbon and yeah, the waiting time is a bummer indeed. I owned LC V1 from cavalli and I am surprised how good that pair was. The bass is tighter and it doesn't bleed to the mids. The treble is amazing because it is pushed a bit forward and the soundstage is a tad bigger. Now that I use a 4 times more expensive amp, I felt like the LC comes quite close when I use HD6XX. Separation, clarity and more pronounced treble are the main improvements but by a small margins. But, the case is different with HD800. HD800 drastically improves with my current amp.

This is assumed that the LCX has the same sound as LC. We never know until we get the real thing. I had bifrost+valhalla2 in the past and they are quite good. Although, instead of buying that, I would suggest you go for modi multibit and LCX. By the way, what is your budget? It is easier to suggest what's good for you if we know what is your budget.
True, I wouldn't want that either. I have tried the HD6XX with mojo and it doesn't suits me. Regarding the CTH amp, I have a thought that it could be warmer than LC since it's a hybrid.