There are 3 types of simple closed curves on a twice punctured torus.

The first type are two isotopy classes of curves which are isotopic to the two boundary components. These are primitive and separating.

The second type is separating curves which cut the torus into a once-punctured torus and a pair of pants. These curves are not primitive, since they are commutators (although I assume this is not what you mean when you observe there is no 1-1 correspondence between primitive curves and simple closed curves).

The third type is non-separating curves which are primitive. These curves have the property that they are sent to their inverse (when keeping track of orientation) under the elliptic involution (which exchanges the two boundary components). So this property is certainly a restriction: a primitive curve conjugate to a simple closed curve which is not isotopic to a boundary component must be sent to its inverse under the elliptic involution. It's possible that this condition is sufficient as well, but I haven't thought it through.

There are 3 types of simple closed curves on a twice punctured torus.

The first type are two isotopy classes of curves which are isotopic to the two boundary components. These are primitive and separating.

The second type is separating curves which cut the torus into a once-punctured torus and a pair of pants. These curves are not primitive, since they are commutators (although I assume this is not what you mean when you observe there is no 1-1 correspondence between primitive curves and simple closed curves).

The third type is non-separating curves which are primitive. These curves have the property that they are sent to their inverse (when keeping track of orientation) under the elliptic involution (which exchanges the two boundary components). So this property is certainly a restriction: a primitive curve conjugate to a simple closed curve which is not isotopic to a boundary component must be sent to its inverse under the elliptic involution. I think this symmetry condition is not sufficient, since the subgroup of $Aut(F_3)$ commuting with this involution is larger than the mapping torus of the twice-punctured torus.