Hockey Nut, welcome to House Repair Talk.
This is a great question, if it were my house I'd find an installer that is qualified to remove the existing siding and install new. Why? you said it, the hard surface siding will crack underneath. IMO you will get a better install if you start fresh from the substrate. Asbestos removal requires a little extra effort, air quality testing, and wrapping of the old material in plastic and proper disposal at a approved landfill. We usually do a couple of asbestos shingle re-roofs each year, it's really not that much of a problem. We have never spiked the meter on the testing box. You create problems when you make dust and have airborne asbestos particles.
...but its your house, hire a installer who you are comfortable with and check out their references, customers, suppliers and don't pay for the install in full till the jobs complete and you are satisfied with their work. Just my