Yes, now I see it and I gotta admit I've never seen it before. Nor have I even heard of hardwood flooring done that way. We live, we learn.
Customarily we would recommend you remove the hardwood and install a layer of nominal 1/2" plywood where you planned your tile installation, but I don't think that's gonna work here. But customarily we'd see a sawn board subfloor where you have only about 2/3rds of a sawn board subfloor, which you're more correctly referring to as furring strips.
If you remove the hardwood flooring, you're correct that you'll need to provide support for your existing furring strips and hardwood unless the cut is made over a joist top. But with your access from below I think it would be easy enough to provide the necessary support wherever you need it.
Does your new tile installation directly meet the old hardwood installation at some point?
Your new minimum nominal 3/4" plywood subfloor would also, of course, be supported at all its edges.
One company, Schluter Systems, makes a tile substrate called
Ditra XL, which they indicate for ceramic tile installations over single layer subflooring over joists with 24" spacing. I wouldn't do it, but they say it works. And your Encaustic tiles are not of the same known structure as are ceramic tiles and that might not be in your favor. But that installation style would keep your new tile only a little higher than your existing hardwood. I think.
I would customarily recommend a second layer of at least 1/2" plywood subflooring, then your tiling substrate and tile. With your thick tiles, that would give you a more pronounced transition to your existing flooring, though.
Hang on and maybe we'll get some more enlightened opinions on this.
My opinion; worth price charged.