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You can use

sage: W = WeylGroup("B2", prefix="s")                                           
sage: w=W.an_element()                                                          
sage: w.to_permutation()                                                        
(2, -1)

And relabel the poset using this map.

You can use

sage: W = WeylGroup("B2", prefix="s")                                           
sage: w=W.an_element()                                                          
sage: w.to_permutation()                                                        
(2, -1)

And relabel the poset using this map.

sage: def new_label(w): 
....:     return ''.join(str(u) for u in w.to_permutation()).replace('-','0')   
sage: W.bruhat_poset().relabel(new_label)                                       
Finite poset containing 8 elements