I would suggest that you work with the Kummer surface $K$ of $J$
instead of using Mumford coordinates. The advantage is that $K$ is
a quartic surface in $\mathbb P^3$; in the case you are
considering when the curve has a unique point at infinity,
the vanishing of the first coordinate means that the point is
in the theta divisor, whereas it is the origin when the first
three coordinates vanish (using the standard Kummer coordinates
as in the book by Cassels and Flynn). Since your endomorphism
commutes with multiplication by $-1$, it induces an endomorphism
of $K$. This will be
given by a quadruple of homogeneous polynomials of some degree $d$
in the four coordinates; it should not be too hard to figure out
what they are from the generic representation in terms of the
Mumford representation. Then your problem comes down to checking
whether the first of these polynomials vanishes, and if so,
whether the next two also vanish. (This assumes that all four
polynomials do not vanish simultaneously at some point on $K$.)

When $\gamma$ is multiplication by 2, for example, the polynomials
are of degree 4 and can be obtained via

    KummerSurface(J)`Delta;

in Magma.