I personally ride with a halter and lead rope, OR you can ride in a halter/bridle combo. A caveson or a tie down sit in the same place as a halter. A caveson is supposed to help keep the horse's mouth from opening too wide.
So how can you justify saying that a halter alters the pressure?
If you stop for a break, and you are removing the bridle and trying to put on a halter... what happens IF your horse spooks.. and then you have nothing ON your horse, or its dragging the leadrope and halter that is around its neck.

Here are some examples of neck ropes and the like
http://www.rods.com/p/7661,887_10-Neck-Rope.html

Quick catch rope/halter
http://www.sunsethalters.com/qchalter.php

Halter/bridle combo
http://www.davewilsonharness.com/Endurance.html

And an english style cotton rope halter.
http://www.raraequus.com/productcart...&IDCategory=10

As I said I always ride with a halter and a lead rope, even english, I do it like a calvary rope, around the neck. You never know when you will need one. You can keep one tied on the back of the saddle via the saddle strings, OR in a saddlebag...
Just a thought.