To solve this problem, we can use quite basic tools.
$1$. Locate the centre $C$ of the circle. This is the midpoint of the line segment you were given.
$2$. Compute the radius $r$ of the circle.
$3$. Which one of the candidate points is at distance $r$ from $C$?
Once you know the coordinates $(c_1,c_2,c_3)$ of the centre $C$, and the radius $r$, then the equation of the sphere with centre $C$, radius $r$ is given by
A candidate point $P$ lies on a circle with the given line segment as a diameter if and only if it lies on the sphere with the above equation.
Or else we can use the Pythagorean Theorem. Find the square of the length of the diameter. For which of your candidate points $P$ is the sum of the squares of the distances from $P$ to the diameter ends equal to the square of the diameter?
Or else we can use perpendicularity directly. Let our given points be $A$ and $B$, and let $P$ be a candidate point. For $P$ to lie on a circle that has $AB$ as a diameter, we need the "dot product" of $A-P$ and $B-P$ to be $0$. That might be fast enough for the $1$ minute restriction.