I would do both, train and buy a halter bridle. My horse has never tried to go any where when I put his bridle on, but I still use a halter bridle. Things happen, even with the best training and the quietest horse, so I really like having the security that I never have to take off the halter.
I just don’t care for the ones that attach the bit to the cheek straps. I have the add-on headstall.