I'm in Wisconsin with 2" nylon meeting over the top. It's been through 3 winters with only one issue. Most snows fall right through without any issue. The 3 times I've had it stick, the snow started wet, so ice and crusty stuff built up on the net, and then continued with wet snow, which built up on top of the ice.
Two of those times I went out once during the storm and shoke it off with a stick. No issues. The third time I didn't and it was a bad storm, the snow built up to the point the net came down (which was actually not a big issue, as it just ripped a tiny hole over the central support post, and then laid on the ground. the perimeter remained attached. I patched the 4 inch hole at the support pole and had the fence back up in 20 mins after the storm passed).
I use 50lb fishing line to provide structure between the support posts, rather than the board you currently have. The line will break in heavy snow load and allow the net to fall down with minimal net damage, and the line will stretch in lighter loads and support the net. If you leave the board up, all the net and snow weight will be on that board, and likely cause bigger damage if it does go down.
2"x2" nylon aviary netting from Amazon. No smaller holes. Get the thicker stuff if you can. I have a 25'x50' section up.