Have you tried other foods than grain free? Why does your vet think one breed would have more benefit than any other. That is illogical. I suspect it is causing multiple excretions
Anyway, dogs of 14 weeks upwards have some control of their motions.
For my young puppies I have a 5 hour regime - for every pup I've ever had and that's a lot !
Times can be varied to have 5 and a half or 6 hours between them, but I never leave the last meal later than 6 pm
Meals at 7 am, 12 noon, 5 pm
Toileting outside a few minutes afterwards.
Play after that, then another toilet break
I rarely have had any night-time accidents.
edit - I have used cages/crates for 40 years with no adverse effect on the pups.