Here is suggested list of items you can take while you are training.
What to take when you are assigned a trainer
I am assuming you will be going on the road with a trainer in their truck so you need to think of it as you are a guest in their home. So to be consideration of their home and space will fall to you as the guest.
I do suggest your luggage be packed in a duffle type of bag because it is easier to deal with then hard luggage in the truck.
Also bringing your own bed linen......in this case perhaps a sleeping bag will be better for you and your trainer.