There is nothing wrong with removing the tube from either the bladder and/or bite valve. If the connection is not tight after reattachment, cut off enough hose so that you have fresh "unstretched" hose to put back over the tapered/stepped nozzle. It will have the same seal as when you originally got it. So you lost an inch or two of hose length. Whatever.

if you take the mouth piece off odds are that connection between the tube and the mouthpiece will leak when you put it back on.

Feeding the brush through the tube can be difficult. Remove the bite valve from the tube and feed the brush through that way. Just give it a good tug; it will come off. Do NOT remove the tube from the bladder. Sometimes the brush can get stuck while you are feeding it through. Notice that the long grey plastic line the brush is attached to is curved from where it was rolled up during packaging. If you bend the tube in the same direction as this curve it will be easier to push through. Otherwise just wiggle it around and eventually it will work it's way through. Good luck. It takes a little practice to get it to work, but the tube will be nice and clean after.