So, I figured it out myself.
The most important thing is to remove the "Copy Rotation" constraint from all tail bones—otherwise the tail will stretch.
To bend it, pose the tail on the first few frames, select all the bones, and press Ctrl + A → Apply Selected as Rest Pose.
You can fine-tune the behavior using the Stiff, Dampen, Gravity, and Mass parameters. I recommend setting gravity to 0, but feel free to experiment.
After baking, if you delete the keyframes, the addon will "freeze" — you can re-enable it again from the addon’s menu.