How can I start a floating TLB from existing TLB?

With the "Auto-sensing" feature you can switch between different toolbars depending on the active (or running) programs. Perhaps this switch could be linked to a hotkey and that would be even closer to your request.

On the other hand, I think sometimes people are too focused on a single solution, while there are countless similar solutions already available. Use your fantasy and play with the various features to see what comes closest to what you want. Really, TLB already is one of the most user-configurable programs I have ever seen...