I can't begin to understand what you deal with in Minn but anything you can do to raise the temp of the basement might help. The fact is that tile will always feel cold without supplemental heat but really good insulation will help.

david_cary is usually right but here is a case where I'll rewrite the concepts a bit.
1./ insulation is only good if you have a heat source, so insulating underneath the tiled floor is only good if there is a heat source between the tiled floor and the new insulation. Without any new heat source, any and all new insulation will only serve to keep the tiles warm IF you laid down on them for hours and warmed them up. Otherwise new insulation will be pointless, and will frustrate you.
2./ basement warming will not end up in the bathroom floor tiles, for all practical purposes, unless you overkill the concept of basement warming. If this needs more explanation I could write a few sentences more.

Hope this helps.

Disagree. Insulation helps maintain a steady state.
Not that I disagree with your assessment.

"Could standard electric underfloor mats be attached between floor joists from basement...?

Short answer, yes. But of course, effectiveness depends on actual construction and materials. Do you have ceramic tile on bedboard on wood subfloor? Iffy.

Any heat you put under the floor will radiate into adjacent material(s), doesn't matter whether water or electric is the transfer medium. If it wasn't for kWH price and demand, electric would be hard to beat.

You would have a substantial lag between start time and comfort with an aggregate of poor radiating materials.

All the above cast doubt on 500 bucks. But thirty sq.ft. isn't a lot of area either.

Yes, floor is ceramic tile on cement board (or whatever they use) on wood subfloor. House was built 1927, if that makes a difference. Walls are also tiled.

Disagree. Insulation helps maintain a steady state. "

Insulation only slows the movement of heat, it does not produce heat or stop the movement.

Without some source of heat all that you will have is an even colder room.
The insulation also prevents heat from moving INTO the room from a warmer area.

Oh? Refute "helps maintain a steady state" please.

the thing brickeyee did was to make it more practical to the end user who has to run with the knowledge obtained.

the mental model is slightly altered when one introduces the notion of heat source.

ta da.
hope you are open minded enough to let it be.

--

one thing I wrote wrong in my first post was not saying the heat source could be above the tiles (or below); any heat applied to the tiles.

Thermal insulation inherently means seperating one temperature from another, thus intending a steady state, whether subject to "heat source" or "cold source".

Exploring clearer language hopefully does not imply closed mindedness.

steady state is the concept that is most rebuttable since insulation slows down heat transfer so it only gives one the impression of a steady state; it does not truly maintain steady state, since it allows heat transfer to occur. (Everything allows heat transfer to occur and nothing stops it).

MAINTAINS a steady state.

"Everything allows heat transfer to occur and nothing stops it"

Yes, but since most of us prefer a steady state, insulation is usually better than none at maintaining it.