For an elliptic curve over any field $K$ of characteristic different from $2$, the Kummer sequence reads

$0 \rightarrow E(K)/2E(K) \stackrel{\iota}{\rightarrow} H^1(K,E[2]) \rightarrow H^1(K,E)[2] \rightarrow 0$.

If $K$ is a finite field, then the Weil-Chatelet group $H^1(K,E) = 0$ (e.g. Exercise 10.6 in Silverman's *Arithmetic of Elliptic Curves*), so $\iota$ is an isomorphism in this case.  Therefore you are trying to show that the class of $P = (c,0)$ in $E(K)/2E(K)$ is $0$ iff $\iota(P) = 0$.  

Moreover, since you have full $2$-torsion, $H^1(K,E[2]) \cong (K^{\times}/K^{\times 2})^2$ and in this case there is a well-known explicit description of the Kummer map: for any point $P = (x,y)$ different from $(e_1,0)$ and $(e_2,0)$,

$\iota(P) = (x-e_1,x - e_2) \pmod{K^{\times 2} \times K^{\times 2}}$: 

see e.g. Proposition X.1.4 of Silverman's book.  The result you want follows immediately from this, taking $P = (c,0)$.