Unless you remove the black tank completely and plumb the toilet into the sewer lines, you want to leave the black shut until it’s full. You need lots of liquid to flush out the solids.
As a weekender, I prefer to keep my grey closed for two major reasons:
1) dumping the grey after the black helps flush down the solids. Remember, you want lots of fluids.
2) even the grey gets a bit of solids (food particles, hair, etc...). Again, lots of liquid helps flush solids.
However, since it’s a seasonal, the grey tank is up to you. If you leave the grey closed, you have to deal with dumping it. If you leave it open, you can leave it open all year and never have to dump. The trade off to less maintenance though is you don’t have that reserve of water to deal with points 1 & 2 above.