I'm a bitless bridle convert. Love mine! So does my horse. I'd never ridden with one and neither had he but we tried it and he acted just as if he had a bit in his mouth...but even more relaxed.
Neck reining was the same,as is stopping,turning etc. I just did everything like I normally do and he responded PERFECTLY!
A word of caution tho, not all horses might act so nice obviously so try it out first in an enclosed safe area until you know how they take the new feel of things and until you feel that you have control.
The bitless with the cross under the jaw straps like Dr. Cooks I think are the best. They are different than a hackamore or sidepull.
There are a few cross unders out there but Dr.Cook is most well known, good quality and with some of the others you just don't know because they are trying to copy Dr. Cooks but may not be the same...made a little different, not as good of quality etc. so do your homework before you buy. Which is what it sounds like you're doing.
If you decide on one let us know what you get and how it works out!