Havent been there but have read a lot of their fishing reports on their website every few weeks. From what I can tell they fish a multiple of rivers such as the Renous and Main Southwest Miramichi etc....
They seem to be on fish and will finid them with canoe runs.
If I were to pick it would be the first week of July and even the second week. Should be fish in all the rivers by then and I've done well the first week of July over the last 25 years. I fish most of the rivers.
Regardless of time of year if the conditions are not good then it doesn't matter when you go. It's a gamble. But normally early July is not into heat waves yet and you can still have good water height.
I was there once about 3 or 4 years ago. A very nice spot with rooms for rent in the main house and a number of cabins you can rent on the same property. They fish a number of rivers, their own private pools, leased pools and some public pool. The guide I had was great but weather and water level weren't on my side. (rain and water murky/high) We still hooked onto fish and if there is fish in the river. I am sure they will find it for you. I suggest you call them on the date you are thinking and they will let you know if it is good time to go or not. They also do canoe run / fishing if that's what you like. I had a great time when I was there. Enjoys & Good Luck !!