I'm fairly familiar with this issue as I worked for an environmental regulatory agency for many years and dealt with asbestos abatement. I've also -- like tens of millions of people -- owned a home with vinyl asbestos tile, and asbestos-containing vinyl flooring. You're right to be cautious, but you also shouldn't be paranoid. From your description, this tile currently poses no hazard...unless you start tearing it up.

Professional abatement could be expensive, and it's not necessary. I would suggest that you nail/screw down a new subfloor right over the existing tile, like 3/8" plywood (underlayment grade with no hidden voids), and then put the flooring you want over it. You could both glue and screw down the new subfloor if you want it to be super solid. If the floor is only mildly uneven, the new subfloor will smooth things out. If it's badly uneven, put down some leveling compound and then the subfloor. Yes, this will raise the floor level a little, and you'll need to take out the tub temporarily to make it easier, but this approach will not be that difficult in the long run.

If you have more questions after reading this, please email me and I'll try to answer your concerns.

Check with your state Department of Environmental Quality about asbestos rules.
Tile is not friable and not a very big hazard at all.

I've been doing research on this for my home. I haven't attempted removal yet but I was reading about the "dry ice method".

You place a block of dry ice on a tile and then it freezes. You whack the tile with a hammer and it shatters. Pick up pieces and bag them. Supposedly nothing drifts into the air.

Is this sensible or crazy?

Shattering tiles generally releases fibers. I suggest you consider something else.

Here is a link that might be useful: Floor tile removal

Dry ice is ony used if the mastic will not release.
Hot water and a 6 ionch drywall knife usually work fine.
the idea is to avoid breaking the material any more than you have to.

Thanks for the thoughts. I think we're going to cover it temporarily with a floating pop together floor of some kind, until we can do the whole subfloor over the top and new floor when we get the money and gumption. I'm still not touching it, but at least I'm not as worried about leaving it in its current state. I definitely have other projects to spend my spare time on! (Now searching for how to most easily remove red-colored shellac off a small hardwood landing without sanding--mask or not, I hate putting little bits of stuff into the air.)

We did what you are doing.
In our library we had old vinyl tiles and we didn't want to touch them. The wood under the tiles was too worn to be refinished. We decided to get a wood-look laminate that clicks together. We put an underlayment down (basically plastic sheeting with a little bit of cushion) and taped that together and then put the laminate on top of that and we are very pleased with it. We really liked this also because we didn't have to nail any of it so we didn't have to worry about the tiles.
Before I put the cushion down I mopped the floor with water, to get rid of any dust or anything also. The tiles were in ok shape, although some where starting to come loose from the wood floor. The tiles themselves were not falling apart, but just coming off whole. We decided to glue those tiles on so they wouldn't shift when we were putting the new floor on. We could have easily popped off all the tiles, I think, however I didn't want to deal with that at all. I felt much more comfortable just leaving them there and not having to worry about it at all! We didn't get them tested, so they may not have asbestos.

(Now searching for how to most easily remove red-colored shellac off a small hardwood landing without sanding--mask or not, I hate putting little bits of stuff into the air.)

Soy-Gel. Great stuff, very safe, very easy to use.. literally just spread a THICK coat on, let it sit for an hour or two, and scrape all the gunk off with a 4" razor scraper. Repeat if needed. I'm using it on my kitchen cabinets now to remove the old shellac/grease/etc and it's working great.

Here is a link that might be useful: Soy-Gel

That IS great stuff. I got some for my husband to clean up an old piece of furniture with layers of industrial paint on it. I was being optimistic--what I'm really removing is old glue & mastic from under old linoleum. I thought I was lucky that it tested OK--no asbestos. Just hours of soaking, scraping, soaking more, more scraping. My way of avoiding making a decision on the asbestos tile in the bathroom . . .

"red-colored shellac off a small hardwood landing without sanding"

Shellac can be removed with denatured alchohol.
You could use Everclear if you want something even safer.