Generally you probably want a barbed hook for the Tenkara. But a LOT depends on your skill with it.
Small hooks are usually about the same, barbless or barbed. The small pressure applied to the fish, also means he will generally not run far, though. In some streams water pressure on the line is enough to cause too much pressure. If the fish is caught on a bone, the hook can break off, or bend easily. After some experimentation, a 1 lb pressure on a size 20 midge dryfly hook will break/bend it. So, the slight bit of extra holding power only means the hook will not be "stuck" if the fish shakes his head, but this is often taken up by any water pressure on the line. In flat water, you could loose the fish.
Larger hooks make more difference. And a lot depends on the type and mfg. And the hooks will vary between batches. I used to always squash down a barb, as needed. Some hooks break the entire tip off. So, I started filing them. I noted a BIG difference between a squashed barb and filed(diamond stone) barb. The bump left by a squashed barb helps hold a hook in place. Filed or true barbless hooks slip out. Both are easy to remove from a fish. Size 10 and above, I almost always squash a barb down. They just are easier to set on a fish.
The number of fish I have lost because of shaking a hook, because it was barbless, is way less than 1%. I would guess, closer to 1 in 300 or so. Unless the fisherman makes a mistake, it is no harder fishing with barbless than barbed hooks, really. Barbless hooks set a heck of a lot easier, so, I assume my catch ratio with nymphs, emergers, and streamers actually increases per number of strikes. Many times I will just suddenly have a fish on I didn't know about.
With a set line length, like the Tenkara, just use a heavier tippet with a squashed down barb tip (if the hook will let you.) If he runs, you will break him off, barbless won't matter. If he is smaller, you will like the easier set and you can easily control him with the longer rod. Others will have different opinions, YMMV.