AD744 is a bit shy on the bass, especially comp pin out. Using the normal output, this opamp is neutral but boring - still bass shy, undynamic, unclear. I have a LISAIII clone (without bass boost) with this setup AD744/829. I've tried it with comp pin out vs standard pin out, and I've tried it with other opamps. AD829 is warmer and less harsh compared to AD744, and I think they're a good blend. The final result is just a little bit leaner than neutral. You can get a warmer sound by rolling opamps to OPA134. I've never heard OPA627 or AD797, but they would probably give you the sound you want. Or why don't you add a bass boost circuit like in PPA or LISAIII and have the best of both worlds - details, soundstage and mucho bass.
I don't think the power section will alter the sound. I've tried adding/changing caps - no difference, going from battery to wallpower to battery - no difference, rolling opamps - major difference, rolling output transistors - audible difference (BD137/138 warmer than MJE243/253). I think it's the active parts (and caps in the audio path if used) that makes all the difference in a 3 ch amp.
I've built a 3 ch 2x9V battery powered amp with AD8599/AD825, BTW utterly neutral, and at first I only used 0.1 uF ceramics from rail to rail in close proximity to the opamps. Later I've tried adding electrolytics - 100 uF Sanyo OS-CON, 330 uF Panasonic FC, 2200 uF ELNA RJH - both from rail to rail and from rails to ground. I've done this "live" while listening, and there's no audible difference adding them! Not to my ears. I have HD650 and use an old H/K CDP and a modded iRiver IHP120 as source. Maybe there would've been an audible difference with better phones and source, but I doubt it would've been siginificant. Power caps = overrated?