Dean DeLeo's tone is something I have studied for a long time and I have been down the rabbit hole and back with the Axe.
The first thing you need to figure out is which tone are you looking for? Crackerman? Interstate Love Song? Hickory Dichotomy? The tones you hear on the records are complex and created with things he rarely uses live. For instance Interstate Love Song was recorded with 2 Telecasters and a Supro with a 10" speaker (
https://www.youtube.com/watch?v=th_GUU-MKpk&list=FLrMY_XqWlJ_JATk-XZraYtw&index=9) There is a ton of layering taking place which can sometimes create an effect of modulation (which Dean says he rarely uses in the studio). He's now using some amps from Todd Snider in the studio as well.
There are also tones like Sex Type Thing that are good in the mix, but questionable when isolated (
https://www.youtube.com/watch?v=y-QTv9aiUJY&list=FLrMY_XqWlJ_JATk-XZraYtw&index=7).
If you ask me, he really doesn't try too hard to copy his recorded tones when playing live with his Les Paul (mainly), Demeter TGP3, VHT Classic, AC30 rig. I have sat directly in front of his rig at several shows to listen to just his tones and it sounds great but the tonal changes are largely coming from the volume knob and his attack. That being said, you can come real close to any of these tones with the Axe just using one amp. I think if you truly wanted to nail the live tones, you would need to run W/D/W with completely different signal paths within the axe, which is not very practical.
For me, I have three DeLeo patches: a medium gain rhythm (Interstate Love Song), heavy (Piece of Pie), and a lead with a touch of autowah (Crackerman, Vasoline, etc.). For the medium gain I am finding the Morgan AC20 works great as it already has a Marshall+Vox thing going on. Just remember to use less gain than you think and add a bit of chorus for the modulation. For the heavy and lead tones I am using TS808Mod in front of the Friedman BE with a bit more bass and presence than usual and I am getting great results in a live setting. I add in just a touch of chorus and keep the gain lower than one would expect to get great note separation and clarity. You can also do a tone match to get you the extra step if necessary.
Rock on.