I tile every day of my life, and pretty much know that an acceptable standard for anything containing asbestos is COVERING/Painting etc. as opposed to a DIY removal and remediation.
In your case, given that it's a basement, I'd stay away from hardwoods or laminates, and go with tile, BUT...
NOT directily on Lino. No offense to any who say it's OK,,, Lino is like Vinyl stock, not at all porous, not at all designed for thinset.
I'd sacrifice 1/2 inch in height initially to lay Concrete backer board. Not only will this aid in covering the Lino, but also a better substrate for tile. Certainly for budget concerns you may not have unlimited resources, but in an end result for anything long term you should consider investing. You can SEAL over the lino first if you feel some sense of more comfort and security. Add a vapor barrier, and assume that you should use exterior mortar and grout.
Obviously too, the concrete backer board will be installed into concrete, so you'll have to allow the proper method.
Once the substrate is secure, then the tile can proceed, and still in total, you lose less than 1.5 inches floor to ceiling, in height.
Just my 2 "sense"