Extra peace of mind is only a 20 buck nylon sling away. Get a 3 foot nylon climbing sling. This only works if your lineman's lanyard has a carabiner on both sides (can't be girth hitched to your saddle on one side). Once on your platform with your tether and lineman's lanyard both still on, sit into the tether and feel that it is holding you, loosen your lineman's lanyard a bit and confirm that everything is at it should be. Next, take out your 3 foot nylon sling and girth hitch it onto the rated belt of your saddle, not the bridge, but off to the side so it doesn't interfere with your belt buckle. Disconnect the left side of your lineman's and immediately clip it onto the other side of the lineman's rope, creating a girth hitch around the tree connected to your right lineman's loop. Next, unclip the remaining side of the lineman's and clip that to your nylon sling. The sling should be on the carabiner of the lineman's that has your adjustment (prusik, ropeman, whatever you use). Slide the girth hitched lineman's lanyard up to right below your tether. Adjust the length so that it is slightly loose and will not bind up or inhibit your movement. I then wrap a nite ize gear tie on the repurposed lineman's lanyard to keep it from loosening from the tree and sliding now. You now have a completely independent safety system. If you want to go one more step. Make yourself a pants belt out of 1" climb rated tubular webbing and a rated 1" Cobra buckle. When you girth hitch your nylon sling to your saddle belt also include as a bundle this homemade riggers belt.
With the system I described, the only way you are falling is if you get in a rush and do something unfortunate. I think it is better than running two bridges because it is more independent. You would need two different things to both break at the same time or one right after the other. The only single point of failure I can think of is if the webbing that comprises your belt and the webbing that comprises the saddle/bridge loops that the webbing belt is sewn to all ripped as a single unit at some point, which is possible but being struck by lightning is probably as likely so long as your saddle is a halfway decent design and in halfway decent shape. If you were to include the extra rigger's belt, then that stops that single point of failure as well.