Square of plywood could be interesting. Have you considered 3/4” osb? Cut into say 24” squares, sand the face well, slight chamfer on the edge, several coats of good clear finish. Seal all 4 edges and both sides with a coat or 2 before installing, add more coats on top for wearability. Countersunk brass screws at each corner. You could use a thin layer of adhesive troweled on with a very small v-notch trowel.

Nothing wrong with running screws through the tiles, you arent going to disturb enough to worry about. If you have sealed the back of the ply or osb, you could give the tile a light squirt of water just before laying the wood, that would keep any drilling dust contained.

 

Square of plywood could be interesting. Have you considered 3/4” osb? Cut into say 24” squares, sand the face well, slight chamfer on the edge, several coats of good clear finish. Seal all 4 edges and both sides with a coat or 2 before installing, add more coats on top for wearability. Countersunk brass screws at each corner. You could use a thin layer of adhesive troweled on with a very small v-notch trowel.

Nothing wrong with running screws through the tiles, you arent going to disturb enough to worry about. If you have sealed the back of the ply or osb, you could give the tile a light squirt of water just before laying the wood, that would keep any drilling dust contained.

i was reading some people apply shaving cream or something thick to their screws to keep things from flying around as they get drilled into the tiles. I feel like there must be a way to do it that would not cause problems upon installation, but if anything ever needed to come out or be replaced I wouldn't want to have made things worse by damaging the tiles. So I am more concerned about the future than the present...

I have considered OSB if i go with painting. I originally thought cutting tiles out of plywood would look good but I am more of a painter than a woodworker so it would be easier to just paint a design rather than cut and stain the wood. I'm not sure if the money saved by going for OSB will be worth the effort it will take to get a smoother surface in the end. It's not a big enough space to make a huge difference in cost but it would be nice to save some money.

 

You can use a spreadsble adhesive like this, just spread only enough to get the work done, it will skin over in a few minutes. This is from ace hardeare, lowes and hd sell the same or similar.

 

Pull the staples the asbestos is dangerous when you make a dust cloud and work in for a few years.
Plywood would need to be nailed down and that could cause more trouble than pulling a few staples I would go for a floating floor and the new vinyl that look like laminate would be what i would go for.
How to Install Luxury Vinyl Plank Flooring - YouTube

 

To be honest i am not too concerned about the asbestos risks since I think it's not that big of a deal when the proper precautions are taken. However my wife is very paranoid about anything asbestos related so I'm trying to be extra careful to ease her stress.

I want to avoid vinyl because i really don't dig the greyish interior design. But also I want to have a paintable surface like plywood because I am a muralist and want to eventually add some hand painted patterns or designs to the floor. It's a farmhouse from the 1800s so ideally I'm going for a more modern version of traditional painted farmhouse floorboards.

 

You can leave asbestos in place as long as you encapsulate it.
A glue down floor over the asbestos tile would accomplish that.
Nailing or screwing will certainly disrupt the tile.
In a commercial setting with concrete floors, I saw a contractor flood the floor and let it sit overnight and scrape up the tiles.
I pulled my crew out until that farce was over and the floor work was over.
I'm not sure a floating floor would satisfy some AHJ.....but you can always ask.

 

I as thinking osb because a clear coat finish, or light stain, brings out the different pieces of wood. Squares of cabinet grade ply would look good, especially if you paint them.

I was thinking 24x24 squares to emulate tiles, or even linoleum squares from when the house was built. Also easier to install smaller pieces, especially cutting in around corners. And full sheets of ply would look like unfinished construction.