The concrete is already fully cured (generally accepted cure time is 28 days) So I don't think you need a few months. A couple of weeks of heat and a cool off and see what happens.
Because the coefficient of heat expansion is very similar in concrete and stone, your main concern is the shrinkage cracks.
I'm sure that one could make arguments for or against the membrane. I have always dealt with fairly green slabs, so I just put it on automatically. I understand the cost issues, but $1500 on membrane is not that huge when compared to the cost of tile, labor and materials to re-do a cracked floor.