N.B. This is an edit of my original post, confirming the guess that I made originally.
The answer is no, i.e., such curves are not forced to be ellipses.
Here is a sketch of the argument. (The details will take a while to type in, and that will have to wait. Also, I'm not sure that there will be that many people interested in seeing the details, so I might not put them in unless I get a request.)
Such a curve $C$, if it is strictly convex (which I am assuming from now on) defines a closed curve in the $5$-dimensional space $T$ of triangles in the plane with fixed perimeter $P>0$. (Note that $T$ is an $S_3$-quotient of a hypersurface in the space of triples of non-collinear points in the plane.) These triangles of perimeter $P$ are so-called 'billiard triangles', i.e., the angle of incidence and reflection of the sides of each vertex of the triangle with the curve $C$ are equal.
This means that such a curve in $T$ is tangent to a certain rank $3$ plane field $D\subset TM$, and, conversely, a closed embedded curve in $T$ that is tangent everywhere to this plane field $D$ represents a closed curve of triangles of perimeter $P$ moving in such a way that the velocity of each vertex is perpendicular to the triangle's angle bisector at that vertex. Thus, the curves we want to study are solutions of an underdetermined system of ODE.
The $D$-integral curves represented by the ellipses (which do give solutions, by Chasles' Theorem) are 'regular' in the sense of control theory (this is what I had to check by looking at the structure equations of $D$), so that one can make an arbitrary functions' worth of perturbations of any such closed $D$-integral curve (say, one given by an ellipse) to get other nearby $D$-integral curves. Thus, there will be lots of such closed convex curves, near ellipses but not ellipses, that have a circle of inscribed triangles of maximum perimeter.
It might be interesting to construct some explicit closed integral curves that don't come from ellipses, but I don't see any easy way to do that right now.
Remark: The plane field $D$ is interesting. It is bracket-generating and does not contain an integrable $2$-plane field, so that it belongs to the type studied by Élie Cartan in his famous "Five Variables" paper of 1910. It is not 'flat' in Cartan's sense, i.e., the group of symmetries of the plane field is not of dimension $14$, but rather of dimension $3$ (very small). Still, the general theory says that the $D$-integral curves near the ellipses (in a suitable sense) are regular curves, so that they are freely deformable.