You can simply take random samples (i.e. submatrices) and estimate the Mantel test statistic. With enough samples, this should converge on the same conclusion. In this way, you can avoid having to cook up a method for accessing all of the data.
However, as these are sparse matrices, it's better to go after a method for accessing the data via coordinate object lists (COO representation), i.e. (row, column, value), and implement a simple calculation for tabulating the test statistic across the two matrices. You can merge the two matrices' COO versions or perform more complex tabulations using the
Ultimately, it depends on how many values exist in your matrices (i.e. how dense they are). Attacking them in sparse matrix formats may be best if they're very sparse. Using a subsampling method may be better if they're not super sparse.