Your concerns for the hazards from asbestos asphalt tile are unwarranted. There are some very easy steps you can take to minimize the possibility of fiber release from the tiles. I strongly recommend you remove the old tile and padding. Wet clean the tile floor using soapy water and a mop so you have a good surface. Remove and dispose of any loose tiles and use a floor leveling compound to fill in voids.
To apply the tack strip, simply apply a strip of duct tape around the perimeter of the room. Tack the strips right through the tape, any dust or fiber will be captured. Apply carpet per usual methods.
Please post back if you have concerns about this, I have dealt with these issues for many years as a regulator and consultant and can assure you that the risks implicit in these recommendations are minimal to none.