I've been drooling over the LPGT. Almost traded my WM1Z for one. A TOTL DAP makes a difference IMO.
You should give the Cayin N6ii a go with T01. I love mine. I believe I like it better than the stock WM1Z. Now with an external amp the WM1Z really shines. The built-in amp in the WM1Z is weak IMO. Everything seems recessed behind the vocals. I'm still working on getting the modded firmware working with my WM1Z. Hopefully this make a difference with the sound signature of the amp.
Regarding the amp section on the WM1Z, I don't know much about its weakness or strength, but the awesome Whitigir has come up with hardware mod plans for the WM1Z, which also include adding capacitors to boost the power of the WM1Z
A company called Music Sanctuary offer their 'K-mod' at 3 different tiers, but I suspect the Whitigir mod will be better.
https://www.head-fi.org/threads/sony-nw-wm1z-wm1a.815841/post-15508505
That's a link, and you can post a question or PM him for more details I guess (if interested)