What are you using for bottom insulation?
I don't see any mention in the OP regarding what bottom insulation is used, if any... IMHO full side-to-side bottom insulation would be a key component in using a TQ in colder temps...
I always wear either silk or wool longs as sleeping clothes - so a possible draft wouldn't be much of a 'shock' or bother to me in that case (another caveat - I'm a relatively warm sleeper and have not been 'cold' sleeping in my hammock in the low teens with a 20* 2/3 UQ & 30* bag as TQ)...
A properly fitted UQ wraps up around you on the sides providing insulation, thus a TQ does not need to 'wrap' around (or under) you very much at all... If your bottom insulation is only a standard width pad (20"), your shoulders and possibly hips/thighs might be without much insulation (or if you are relying upon the down in a bag or TQ and it gets compressed, you're back to square one dealing with CBS)...
I still only use my WM bags as TQs in my hammock - but a Burrow TQ is definitely on my Christmas wish list!
The TQ will be much lighter than a SB since it won't have all that excess 'unneeded' fabric & down (and also would be much easier to 'handle' inside the hammock, too)...
As for tossing & turning in my sleep - I do when trying to sleep in a bed or on the ground, but I do NOT when sleeping in my hammock! ;)
Take a look at your bottom insulation and perhaps 'experiment' a little more - then, if you still want to ditch your TQ, let me know! :laugh:
Happy hanging!!