Brian,

I'm just going to think about the case when J is purely toric because
it suffices for my application and I can use "Tate curves".  I'm not
sure what is going on in the general case.

CLAIM:  Let 
            0 --> A --> J --> B --> 0 
be an exact sequence of abelian varieties over K=Qp^ur whose Neron 
models have purely toric reduction at p.  Then J(K)-->B(K) is surjective.

Proof.  This is an application of Hilbert's theorem 90 to the Tate
uniformization of A, J, and B.

 Consider the Tate uniformizations:

         0       0       0
         |       |       |
        X_A --> X_J --> X_B      
         |       |       |
        T_A --> T_J --> T_B      
         |       |       |
  0 -->  A  -->  J  -->  B  --> 0
         |       |       |
         0       0       0

The maps T_A --> T_J, etc. are induced by the maps A-->J-->B. 
The kernel of X_A --> X_J is trivial, because X_A and X_J are free
abelian groups and the rank of the image of X_A is dim(A).
Thus ker(T_A-->T_J) = 1.   By the snake lemma, the discrete cokernel
of X_J-->X_B is the image of the connected abelian variety A;
hence it is trivial.  Consequently T_J-->T_B is surjective (this 
is also "clear" without the snake lemma, by a dimension count). 
We thus have an exact sequence of tori:

               0 --> T_A --> T_J --> T_B --> 0

and a corresponding exact sequence of Kbar-modules:

             0 --> T_A(Kbar) --> T_J(Kbar) --> T_B(Kbar) --> 0.

The corresponding long exact sequence of Gal(Kbar/K)-cohomology yields
the exact sequence

             0 --> T_A(K) --> T_J(K) --> T_B(K) --> H^1(Kbar/K, T_A(Kbar)).

But T_A(Kbar) is a product of copies of Kbar^*, so by Hilbert's
Theorem 90, we have H^1(Kbar/K, T_A(Kbar)) = 0.  We now have a square

                              T_J(K) --> T_B(K)
                               |           |
                              \|/         \|/
                               J(K)  -->  B(K)

where the top map is surjective, as are the two vertical maps.  It
follows that the bottom map is surjective, as was claimed.

-------

What do you think?