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 $(a,0)$ and $(b,0)$,

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

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

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

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

In particular, $\iota$ is an injection. Therefore $P \in 2E(K) \iff$ the image $[P]$ of $P$ in $E(K)/2E(K)$ is equal to zero $\iff \iota([P]) = 0$.

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 $(a,0)$ and $(b,0)$,

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

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

Note that, as Bjorn points out in his nice answer to the question, the finiteness of $K$ is not needed or used here. In my original version of this answer, I mentioned the fact that $K$ finite implies $H^1(K,E) = 0$ -- it seemed like it could be helpful! -- but the argument does not in fact use the surjectivity of $\iota$, so is valid over any field of characteristic different from $2$ over which $E$ has full $2$-torsion.

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 $P = (c,0)$.

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 $(a,0)$ and $(b,0)$,

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

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