I think there are two schools of thought for that...
1) Wearing the old school's patches to a new school is disrespectful, when you leave one school and join another, you need to dedicate yourself to the new school and represent their patches accordingly.
2) Your gi patches are your BJJ heritage. When you leave a school, they have left a mark on you, so it is fitting that they leave a mark on your gi. Your old school and your new school have both affected your BJJ talents, so both should be recognized. Sort of like respecting your high school class ring the same as your college ring. One gym took you so far in your training, and the new school will take you further on your journey.
I for sure wouldn't cut my patches off just for a trial week at a new gym, just like I wouldn't sew on the NEW gym's patches just for a week trial.
New gyms respect other gyms (usually) enough to recognize their belt levels (i.e. a purple will stay a purple even if he moves), so in my opinion, they should respect the training and loyalty that a player has for their old gym.
Be loyal to your schools, but also regonize that if you have trained at multiple schools, they both deserve your loyalty. A gym is like a family. If I ever have to move and leave my gym, it will be important to me to keep my patches on my gi to represent my roots.