How about taking off the tarp and covering the bed with 3-4 inches of mulch?

In winter, in a freezing climate, it's probably not going to make much of a difference. If you're concerned about it you could put something under there at regular intervals to prop up the tarp an inch or two. Wood stakes, sticks or small logs, bricks, rocks, scrap lumber, etc. will all work. I do agree with darth that an organic mulch would be good here, there's no real need for a tarp and nature doesn't use them. :-]

Where in the United States are you? That may determine what is good advice for that planting bed.

I'm in the Chicago area

In the Chicago area at this time of year there will not be much air exchange in the soil because the Soil Food Web is not very active, so the tarp is not going to create an anaerobic environment, now. However, a better cover would be a mulch, or maybe a cover crop, to protect the soil from the ravages of the winter weather. Both would add organic matter that the tarp will not.