Fab geetar - absolutely gorgeous :-)
I was looking out for one of these a while back and then I found a limited run 2016 version with a plain top. Incredibly fabulous guitars. Mine came with 57+/57 Classics.
I've put Stormy Mondays in mine. I believe they'll do what you want.
Haven't recorded with that one yet, but I also have Stormies in a 2012 Traditional that I have done a few recordings with.
All the guitars on the following three are that Traditional with Stormy Mondays. In each song, there was very little changing the amp settings, I just used the controls on the guitar to change the sounds for each part.
This one's got the most lead in it, including harmony stuff near the end, but it's full of "clean" rhythm and some crunchy stuff:
https://alonetone.com/andyr/tracks/silver-swanThis one is blues/rock, I was going for an earlier Billy Gibbons vibe, could've done with a slightly cleaner basic amp:
https://alonetone.com/andyr/tracks/are-you-doneThis one is, I dunno, pop-rock. It's the first time I recorded with the Stormies (I was AMAZED at what they could do ... and as a confirmed strat-man of 30-odd years I haven't really played a strat since!!):
https://alonetone.com/andyr/tracks/dancing-girlMules can do it it too - this thing has a lead guitar that starts in the middle somewhere that I played on Mules in a 335 (the Riff Raffed SG broke a string! I think most of the rhythm gits are the SG and an Explorer with 57s - there might even be some Mississippi Queens in there too - I can't remember):
https://alonetone.com/andyr/tracks/meet-me-at-the-crossroadsAnd this one is all Mules (the same set) in a Tokai Loverock before it gave them up to the 335:
https://alonetone.com/andyr/tracks/muletidePersonally, I don't find the Mule cleans feel so sweet and yummy to play as the Stormies do. They do do clean, but just not quite as organic an vocal as the Stormies seem to be for what I want. This means they don't seem to work so well as "all-rounders" for me. Not sure about Abraxas - haven't tried them, though they are on my list, I suspect they might be a little hot for what you want?
So, for what you're describing, I'd suggest Stormies. Or conceivably a Mule bridge with a Stormy neck? I love the middle position on SM/SM though. And I use bridge pickup clean just as much as I do the neck.