They don't cook well together as they don't cook for the same length of time. Here's a page I posted on another food forum last year, it's a professional resturant forum but it's still the way I've always done my corned beef and cabbage St. Pat's dinner.
The problem I see in your plan is the dish has the possiability of easily becoming a mushy mess. Our corned beef has a large following, in fact, it went on the menu last Thursday and will stay there thru St. Pat's. Here's what we do. Simmer the corned beef with the packets of spices, water (save the beer) barely covering, until tender. I do 10-12 at a time, working one day out. I set it off the stove and let the briskets cool in the brine, adding a couple scoops of ice after a awhile to bring the temperture down so I can set in the bottom of the cooler. I think chilling in the cooking liquid is one of the most important steps.
The next day, remove the corned beef (save some broth) maybe trim the fat up a little, set on a cookie sheet. I make a glaze of a little apricot jam, brown sugar, and lots of brown mustard, lightly coat the tops and bake at 350 till the glaze takes. Let this cool down and slice, and stage.
I cook a batch small whole onions in the broth and remove; cook my carrots, remove, and then russets cuts in half length wise, baby reds or fingerlings. I lay these all in a flat pan, and chill.
I keep a low flat skillet with a 1/2 to 1 inch of broth in it; the sliced corned beef, the veggies are heated to order. As I lift out and plate with a slotted spatula, I turn up the heat and throw in a handleful of thinly sliced cabbage for a quick saute` (I add a little caraway seed) The cabbage stays nice, doesn't look limp and soggy.
We drizzle a little parsley mustard sauce* on the corned beef and serve. Add a small ramkin of the hot broth to the plate. The parsley mustard sauce is a favorite, adds a lot of contrast to the plate visualy and to the taste.
Everything is crisp and clean looking, and the sauteed cabbage looks smart on the plate. Hits the menu at $16.00 with a salad. Now, drink the beer!
*Parsley Sauce: sauteed shallots or onions, add heavy cream, lots of djion mustard, salt, pepper cooked very thick then add a couple of tablespoons of fresh parsly)
Left over corned beef? Rubens!! We had a little bit of Hot Chile Garlic Sauce to the 1000 island dressing, adds a little kick to the sandwich.