The self contained version of this question is: "Given a Möbius strip $M$ do there exist two curves $\gamma_1,\gamma_2$ in $M$ such that $M\setminus\gamma_1$, $M\setminus\gamma_2$ are orientable and with different area?". The area form on $M\setminus\gamma_i$ is the natural one as a submanifold of $\mathbb R^3$. For more context see below.
I was trying to answer this question with a response along the lines of "there is no global area form on a Möbius strip, and therefore no way to define a total area". The obvious objection is that if I take a slip of paper with a certain area, the total area along which one can move on the slip of paper is simply the area of the front of the slip plus the area of the back. This is essentially the notion that OP suggests in asking whether the Möbius strip will have area $4\pi$.
I wanted to make the further counter-argument that this is not an area in the normal sense, since we are actually adding areas of a pair of orientable submanifolds of our choosing to obtain a number which we'd call the "total area" of the Möbius strip (in fact we are adding the area of the same submanifold twice since a mathematical strip does not have thickness, but this doesn't change the underlying question). My differential geometry is a bit rusted, so I consulted the internet myself to be able to give a more precise answer. I eventually found exactly what I was looking for here. So now I only needed to concoct a curve along which to cut the strip so as to get an area different than $2\cdot(\text{one side of the square without identifying})$. But I'm stumped!
Just to close the circle, I think the best answer (from a differential-geometric point of view) is one that Ted Shifrin mentioned in comments. A nonorientable manifold like the Möbius strip has a perfectly good notion of area; you just can't compute it using differential forms. Instead, what's needed is a density, which can be integrated on any manifold. For a manifold (or manifold with boundary) with a Riemannian metric (such as any submanifold of $\mathbb R^n$ with the induced metric), the appropriate object is the Riemannian density, which is the unique positive density that yields the value $1$ whenever it's applied to an orthonormal basis. This is explained in my Introduction to Smooth Manifolds (2nd ed.), pp. 427-434.