Yes the BMCO9 fits your 810-3330S. Here is a link to the parts for your grill - https://grillpartssearch.com/category/Parts-for-BRINKMANN-810-3330-S-Gas-Grill.html
No, this does not fit your grill. Here is a link to the correct parts for your model number. https://grillpartssearch.com/category/Parts-for-BACKYARD-CLASSIC-GR3055-014684-Gas-Grill.html