I haven't made that recipe with red, but the kind we have here is stiffer than green, and I don't know that the texture would be nice, though the sugar and the hot dressing might help. You could make one leaf's worth of shredded red cabbage to try it out. Remember that the dressing will likely turn purple-ish.

Something that's popular here is Firecracker Slaw. There are many different versions, but it usually has red cabbage (sometimes plus some green), thin strings of all colors of bell peppers (green, red, yellow, orange, purple, or whichever are available at the time) as well as carrots (rainbow if you can get them), onion or scallions, and a chili pepper. The dressing is usually a vinaigrette type, but some people do a traditional creamy cole slaw dressing. Which still can turn a little pinky, but I think the hot dressing would bleed it a lot more. The peppers, et al., are at least half the bulk, and soft enough that the heavier crunch of the red cabbage isn't an issue. If yours is tough, however, you could blanch it first, before you cut it. I'd think it would bleed less on the finished salad that way.

IMO red and green cabbage are not readily interchangeable. Red is a much tougher leaf - just think about how long it takes to properly braise red cabbage. I don't care for raw red cabbage (though I love it cooked) and don't mind raw green cabbage (especially napa or savoy cabbages). I will tolerate a little very, very finely shredded red cabbage in a slaw for color, but would prefer to avoid it altogether. So, no, I would not simply substitute red for green cabbage in any slaw, even one with a hot dressing.

I have never made Fred Harvey's with red cabbage, but I often make a green salad with iceberg lettuce, spinach shredded red cabbage ( on the 1 mm shredding disc on the cuisinart) and shredded carrots and dressed with a simple vinaigrette....which is pretty well what the Fred Harvey's dressing is. It is very well liked and the cabbage isn't tough of hard....no tougher than iceberg lettuce...the mid rib is tough but the leaves tender.
Go for it!! It would be a lovely purple thing....and if you shred the cabbage fine enough and marinate more than 4 hours ( I usually marinate over night...) It shoul,d soften up just fine....Just don't put any red cabbage in slaw for color or the whole mess will turn a bluish purple...either go all red or none. One time I put a little "just for some color" into my vegetable pasta salad.....bad idea...stained the pasta, the cukes the zucchini ....also calamata olives will stain. Don't put either into anything unless you plan to eat it very soon.
An aside, My son is having a big BBQ party tomorrow for 50....smoked all sorts of stuff....and I will be making Fred harvery's for all!....taking my 1 mm disc with me....they have the machine but that 1 mm disc is hard to come by!

Anything that any Fred Harvey restaurant ever served was always delicious.

Thank you all for your replies. I made a trial slaw for myself and since it was good I made one from a whole medium cabbage for the office party.

After cutting the cabbage I sprinkled half a cup sugar, mixed it well and left it for half an hour, then grated the onion, mixed it again, and poured the hot sauce.

The outcome is not as tasty as with white cabbage - but a lot crunchier.

Next time I'm going to mix half white and half red, so it'll be both tasty and crunchy.

I made a similar salad with some red and some white cabbage (it was in a pre sliced slaw mix) but it turned out "pink" from the red cabbage sitting. It was pretty just didn't look like normal. So be prepared for "pink".

I'm not sure if this would work, but if you did the Fred Harvey with each color of cabbage in its own bowl and chilled them before mixing them together, there might be less pinkness. ;)

Martha, you're right. How come it didn't occur to me?

Plllog, the red cabbage produces sort of a pink juice when marinated in the sauce for a while. The combination will end up pink all the same, but nice try :)

I think in future I'll just stick to green cabbage. This time when I went to buy cabbage, the green ones were wilted and vile and the red ones were crackling fresh.