Homomorphism of Legendre curve

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.