So, a few things.
First, the Passport is an excellent small and portable chart for field use.
But the DNG profiling software simultaneously sucks and blows, and that's not a compliment.
The first half of the problem is the DNG profiling software's fault: it only uses the classic 24 patches when building profiles; it completely ignores those other 26 patches. Considering that those other patches include both some much-more-saturated patches that really help define the camera's response as well as a number of near-neutral patches that really help nail down where the neutral axis (and therefore white balance) lies, that omission is both incomprehensible and inexcusable.
The second half is Adobe's fault, and that's the incoherent mess that is the DNG color "profiling" process. It's not even remotely theoretically possible to get anything vaguely resembling colorimetric accuracy from DNG profiles, though you can fake it by manually tweaking a DNG profile to be as close as you can get it and then build an ICC profile on top of it. Basically, if your idea of "good" or "pleasing" color means no-holds-barred impressionistic interpretations of color, DNG is for you. But if you want an accurate representation of color, DNG is a cruel and unusual tool for extraordinary rendition.
If you care about accurate color, you simply cannot use Adobe products to develop your raw files. Nor can you use many of the other popular raw development engines, such as Canon's DPP, because they suffer from the same root problem: the programmers have decided that their taste in color palettes is best for you. Accurate color, hell -- you can't even stop any of these raw developers from applying a contrast-boosting (and detail-obliterating) S-curve!
It's a real shame, too, because the hardware itself is quite readily capable of superbly accurate color reproduction. There's no reason why accurate color shouldn't be the default starting position, always available as an option, with the various "secret sauce" recipes only optionally added on top.
So what you're left with is mostly tools that come from the Free / Open Source software crowd, some of which produce superlative results but none of which have user interfaces with the spit-n-polish that Adobe products have. That is, you can use Adobe (etc.) products which are beautiful to look at but which mangle your own images, or you can use other tools that are ugly to look at but which make your own images shine.
If you go that route, you'll want something which is at least loosely based on dcraw for its development engine. My own runaway favorite is Raw Photo Processor, but there are other good options.
And your basic workflow would be to first create high quality quasi-generic ICC profiles in carefully controlled situations. The ColorChecker Passport, as useful as it is in the field, really doesn't have enough patches for that kind of work. There are other charts available for purchase that are usable, but I personally made my own chart; it has a replica of the classic 24 ColorChecker patches, another couple dozen paints, a black trap, some PTFE thread tape, a dozen or so wood chips, and a couple hundred patches printed on an iPF8100. I have some plans for a second version, but it's served me well.
Of course, you'll need a spectrophotometer to build the necessary reference files for any chart; there's enough batch-to-batch variation with any manufacturing process that you'll want to measure your actual chart, even if you buy it from a reputable source. The i1 Pro is an excellent tool for this purpose.
You'll also need software that lets you build these kinds of profiles. X-Rite doesn't include that with the software they bundle with their consumer-level instruments, but they do sell some very good and very expensive software that would work. However, if you're not afraid of the command line, ArgyllCMS produces absolutely amazing results, is free, and is superbly supported by the author on a mailing list.
So, you'd build profiles for each camera with the light sources you most care about; one for each of the camera's pre-canned white balance settings is a pretty good idea. And you might even want to consider building one such profile for each lens, as different lenses have different color characteristics.
Then, when you're shooting, you'd include a shot of the Passport as usual. When you get back to the studio, you'd do a linear gamma UNIWB development of that shot -- basically, just dump the raw file completely unmodified to a TIFF. You'd build a matrix profile from that and do a reverse lookup of D50 white, which will tell you what per-channel multipliers you need both for white balance and to normalize exposure. You'd then use those figures when developing the real shots and apply your most-appropriate custom-built pre-canned ICC profile.
That'll get you as close to perfect color as you're going to get with a DSLR. And, indeed, said color is good enough that, if you've got a similar workflow at the printing end of things, you can make copies of artwork such that the artist herself has to stare a long time at the original and copy side-by-side to be able to spot the differences -- the gamuts of the original and your printer permitting, of course.
One other note...you're having problems with reds and you're mentioning problems with overexposure. Some cameras, especially older ones, are notorious for overexposing reds, especially reds rich in infrared such as flower petals in sunlight. It's impossible to recover an overexposed image, no matter how good your profiling software.
What you can do, however, is underexpose the image sufficiently to prevent the reds from blowing, and still use the Passport to determine how to normalize the white balance and exposure in post. You're essentially applying digital ISO boost in post-production at that point so you have to be careful of noise, but you can boost a 5DIII ISO 100 exposure by a half-dozen stops in post if you're careful so it's not as much of a concern as it used to be.
Good luck, and may the Farce be with you....