My calculator only supports a 3x3, 3x2, 3x1, 2x3, 2x2, 2x1, 1x3, 1x2, 1x1 matrices. Is there any chance I can get the inverse of the 4x4 using my calculator using the matrix mode?
In principle, you could partition your matrix into $2 \times 2$ blocks (i.e. a $2 \times 2$ matrix whose elements are $2 \times 2$ matrices) and then apply "Block Inversion" : have a look at this article.
But unfortunately the resulting formula is somewhat complicated. However it is instructive to know about the division of a matrix into blocks.
I don't think you can find $4×4$ matrix inverse in your calculator unless you find any property which reduces $4×4$ matrix into $3×3$. And, I don't think if there is any property exist.