Shoshone is probably the best bet on the Colorado for still having water, short but I think the rafting trips just lap it. Might be able to head over to the Arkansas as well if you wanted more of a drive and would offer a longer trip if they still have a decent flow, Browns or Numbers seem to fit your bill.
Shoshone will likely be the best CL III choice within a reasonable drive. The Ark will be well past flow augmentation, with only natural flows in the 300-400 +/- range likely. While scraping down Browns at 400 is considered by some to qualify as rafting, my opinion of that run at low water is that it provides a very poor quality whitewater experience.
At least one outfitter in the Glenwood Springs area offers a "combo" trip of rafting on Shoshone (~3 miles of CL III), followed by individual IK's into town (~8 miles of CL II) more or less on your own. You could try to rent a kayak and follow the group, but most outfitters are uptight about non-paying customers riding in their buses for insurance reasons.
My wife and her friend once paid to raft laps on Shoshone with an outfitter while I solo kayaked alongside the outfitter group, but they were not willing to allow me to ride the bus with them on the shuttle when we booked the trip. However, when we showed up my wife's friend flirted with the guide on their boat and he let me ride the bus with them on the laps so it worked ok. You may also consider a run on the lower Roaring Fork (~6 miles of CL II, Cemetery rapid CL III-) with a rented IK and rented kayak combo but I don't know who in GWS rents IK's outright. Any other options would require a lot more drive time with minimal whitewater, with the only exception being if you booked a one day outfitter trip on Westwater (3 hour one way drive to Junction) in which case you would not be able to kayak alongside due to permit issues. Later.
The class III section on shoshone is only about 1mi so with a kayak you can just carry your boat back up to lap it. Blue sky adventures near the hot springs will rent out IKs outright (ie you don't have to go with their trip), but you're not allowed to take them on Shoshone. You could always lap shoshone a couple of times then meet your wife with the rented IK at Grizzly Creek (1mi down) and run the CLII section from there. Otherwise find a outfitter to raft laps on Shoshone and just carry your boat if you can't ride the shuttle with them.
Alpine quest sports is the place for kayak rentals in GW.
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