I don't have a BMBH but pictures show enough to know what the essential problems are.
There are three pieces of webbing coming together at a ring. Two of them are part of the suspension triangle, one of them goes to the tree. You can leave the suspension triangle ones alone, they should stay dry. You'll need to un-attach the other bit of webbing, e.g., cut the stitching.
At this point you have many options. The cleanest one is to loop the tree-webbing piece through the ring, and fasten to itself using a JRB tri-glide. A bit more radical would be to transform the long piece of webbing to tree into a short piece just long enough to protect the tree, and work in a lightweight suspension line between tree and hammock. On the hammock end you'd run the cord through the ring, and tie off with a trucker's hitch or even a simple (but somewhat harder to adjust) round turn with 2 half-hitches.
just my $0.05 worth.