I'm trying to find out what's possible without a tripod. I'd rather not use a tripod if I can get away without one.
I don't think that handholding a TS-E will be effective for panoramic work. I can't imagine how you would shift the lens and handhold with enough precision to get the benefits of the TS lens. Controlling the knobs and maintaining a stationary position would be next to impossible. If you are planning to handhold and pan the camera, I would recommend a normal wide angle lens instead. TS-E can be fun as a handheld lens, but only for single-frame work using the tilt for focus effects and the shift for angled shots (like mirrors and stuff). Precision work with a TS-E demands a tripod.
One problem I have noticed is that a shift of the lens seems to move the entrance pupil (sometimes called the nodal point) of the lens or to distort the image in ways that make stitching nearly impossible. So if you shoot a bottom row at -5mm and a top row at +10 mm you may have parallax problems or simple misalignment problems in one row or the other. This isn't a problem if you are only shifting to make your pano. It is only a problem if you are mixing shifting with panning to make a pano.
Let me try to explain:
If you shoot two shots, one above the other, and you use the shift to offset the frames and make a small pano, you will generally have no troubles. You will have two rows of one image (2x1). Where the problem comes is when you try to do two rows of two or more images. For reasons I haven't fully explored yet, shifted pairs of images will flat stitch perfectly, but adjacent vertical pairs may not stitch well because the upper row (+10mm shift, for example) will have too much error between frames.
I am not positive that the problem is actually parallax. My hypothesis is that the lens projection distorts the +10mm image so much that the upper row corners are warped in ways that exceed the abilities of the stitcher. The reason for this hypothesis is the observation that adjacent shifted image pairs may have a have a tree repeated in both frames but that bend in dramatically different directions (say 30 degrees different).
What does this mean in practice? I found that I could generally shift no more than 12mm from the bottom row to the top row while combining a flat stitch shift pano and a panning ptstitcher pano. Adding a tilt into this equation made the process even harder and meant that I couldn't shift more than total of 10mm.
But as for handheld panos using the TS-E shift mechanism: I wouldn't expect it to work well.