I'm really confused by your reply????? Can you please ellaborate
Capitalization makes a difference.
If you say "catholic" you mean "universal." If you say "Catholic" you mean the Church of Rome. If you say "orthadox" you mean closely following some set of rules or ideals. If you say "Orthadox" you're talking about a church from Greece, Russia, etc.
If you say "Dressage" you mean a very specific discipline with very specific rules.
If you say "dressage" you mean a fairly broad type of training.
Some folks might consider this hair splitting but the devotees of Dressage do not. For them (and the USDF and the FEI) Dressage means "walk, trot, canter."
So, we have to go back to the original question. What does your gf want to do? Training a TWH to trot so that they can compete in Dressage seems to me to be violating a cardinal rule of horsemanship, the rule of "Horses for Courses." If you want to compete in Dressage then get a horse that is suited to that discipline. You won't ruin a TWH if you teach it to trot; indeed that can have very positive influence on gait quality. But no matter how well the TWH trots it's not going to get the kind of scores you need to really be competitive. If all the gf wants to do is "dabble" then I don't see any harm in it (besides spending a lot of time).
If the gf just wants to use "dressage techniques" they she should find an instructor willing to work with her.
Not everyone agrees with me on this, but I think USDF and FEI would. That's good enough for me!