I suspect the answer is that there is no bound on the area -- at least not without restrictions on the metric $g$. My reasoning comes from the following paper which shows that on any three-manifold there is an open set of metrics which admit a sequence of embedded minimal $S^2$s with unbounded area.

In particular, if your manifold $M$ is three-dimensional, then you can modify the metric $g$ in a geodesically convex ball so that the above construction holds inside the ball and so find null homotopic minimal spheres with arbitrarily large area (inside the ball). One can then do a connect sum with a representative of your class.

The intuition for the construction in the paper I linked to (which I admit I haven't read carefully) is, I believe, as follows: think about a surface with two ``pegs" (i.e. two distinct regions that look like large spheres connect summed to an ambient space by thin necks. One should be able to make long closed geodesics by wrapping back and forth around the pegs.

I suspect the answer is that there is no bound on the area -- at least not without restrictions on the metric $g$. My reasoning comes from the following paper which shows that on any three-manifold there is an open set of metrics which admit a sequence of embedded minimal $S^2$s with unbounded area.

In particular, if your manifold $M$ is three-dimensional, then you can modify the metric $g$ in a geodesically convex ball so that the above construction holds inside the ball and so find null homotopic minimal spheres with arbitrarily large area (inside the ball). One can then do a connect sum with a representative of your class.

The intuition for the construction in the paper I linked to (which I admit I haven't read carefully) is, I believe, as follows: think about a surface with two ``pegs" (i.e. two distinct regions that look like large spheres connect summed to the original space by thin necks. One should be able to make long closed geodesics by wrapping back and forth around the pegs.