The local building code mandates borate pressure treated lumber for all framing, so this is all the home centers sell.
There are other sources of wood than home centres! If you go to a dedicated lumber yard not only will you generally find better wood of any given type you'll be paying lower prices for it. In addition I'd be shocked if you don't find a better selection1, which could render the main part of your Question moot :-)
Is it ok to build the base structure out of the available treated 2x6s and 2x12s? Or should I stick to the untreated 2x4s?
Yes. There are many workbenches out there with the understructure, and occasionally the entire workbench, made out of PT wood of various kinds.
If I do opt for treated lumber, how long do I need to let it dry to equilibrate for 70% relative humidity?
This should be a separate Question, because it's a separate question. But coincidentally I tackled this in an Answer just a few days ago.
From the Comments:
Hopefully the PT wood takes up the glue well enough. I worry because sometimes the borated wood has a waxy feeling on the surface.
If you are forced to use the borated wood if in doubt switch glues, because from reading around online the borax treatment does interfere with the bond formation of PVAs. You could try cleaning the joint surfaces well to lessen the effect, or you could not use glue at all....
There are plenty of workbenches out there where some of the major joints aren't glued. In addition to one of the old-school ways of doing the major leg/stretcher joint, a tusked tenon2, you could peg or drawbore all your joints; pegged correctly or drawbored M&T joints don't require glue.
And of course there are always nuts and bolts or lag screws to consider. Again, plenty of workbenches use some combination of these in their construction (especially for the major joints). Modern construction screws such as GRKs are said to be good enough to replace lag screws, with significantly easier installation, in case you want to look into that (note they may be several times more expensive).
I was thinking of laminating 3 layers of 2x4 or 2x6 to make the legs. This way I can get a free mortise for the stretchers and a little extra mass.
Sound plan3. And if you're doing this you could more easily create a tapered mortise for yet another option for a no-glue stretcher/leg joint using an external wedge, see Sturdy, knock-down workbench on Fine Woodworking (PDF available).
1 A quick Google search located one which lists DF from 2x4 up to 2x12.......
2 One of the notable design details of what we've come to know as the Moravian bench. Paired wedges (AKA folding wedges) can be used to give the same effect.
3 Laminating up for major leg structures can have numerous advantages, one of the major ones for the workbench builder being 'free' mortises as you note. But there are often major savings to be had as well, because in wood pricing generally there are premiums for thickness as well as length (do note this does not always extend to width, allowing the possibility of further savings by buying wider boards such as 2x12s and ripping down to width).