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Root finding of polynomials over function fields seems to be implemented, so this should work:

sage: M.change_ring(R.fraction_field()).characteristic_polynomial().roots()
[(x0 + x1, 1), (x0 - x1, 1)]

It has the advantage that it should also work in positive characteristic (where SR would not), and hopefully good use is made of specifics of the coefficient domain to provide efficient arithmetic.