So my GE radio has an S11 base incandescent bulb in it, the original GE one. It's 40 watts, and I want to replace it with an LED bulb to prevent overheating and damage, and save the original bulb from burning out.

I thought of getting an LED S11 40 watt equivalent bulb at Lowes, like this. But all of the LED bulbs I see say not to use on a dimmer. Will they work instead on a high/low switch?

https://www.lowes.com/pd/GE-40-Watt-EQ-2-5-in-Warm-White-Appliance-Light-Bulb/1000247529?cm_mmc=shp-_-c-_-prd-_-lit-_-google-_-lia-_-143-_-lightbulbs-_-1000247529-_-0&placeholder=null&gclid=CjwKCAjw2Jb7BRBHEiwAXTR4jc_ta6fvrkm9z5cR7yvNb04yeuf_JyQBCRhYhLUc8eIjHM05xAuC3hoCjGkQAvD_BwE&gclsrc=aw.ds

The low setting could be implemented in few ways. It could be the same as dimmer (just with 1 set brightness level for the low setting), diode letting through half wave, series impedance (of either inductive or capacitive type), and even actual partial voltage (using a tap from the transformer)

LED lamps come with many types of internal drivers as well

So you get many possible combinations, which may or may not work correctly / work at all / cause damage. It might or might not work even with lamps rated for dimmer use - as their design assumes only standard dimmer (and even then there are 2 options, leading edge or trailing edge)

If you can get more info about how is the "low" setting implemented, this can help

At the very least, if you can confirm that at the "high" setting there is really full line voltage connected straight to the lamp, and the switching is done mechanically (by a relay), then you might get away with using whatever LED lamp at the "high" setting only, and never going to the "low" setting, not even momentarily

In any case, use a LED lamp with no significant capacitance, to minimize wear of the switch or relay contacts from arcing at switch on

What radio is it?

Because with some sets a lightbulb was actually used to drop the voltage for the radio circuit itself (instead of a power transformer). Then you can not go without the real incandescent (or an equivalent resistor replacement)...

It just a modern GE Spacemaker (model 7-5400) one with CD and MP3 dock. It does have an internal transformer I can tell, the rest I don't know about. There a slider switch on the side toward the back of it with off-hi-low. I know sometimes these bulbs blow out and the units still work as far as I know.

I'm thinking maybe just stick with the incandescent. Part of the problem with trying to find a dimmable S11 bulb is these lamps might not be widely used where that would be needed. I've actually seen on some incandescent S11 replacement bulbs that they are used in scent warmers and such - where the heat is actually desired, so an LED version would be useless.

So what remains is to find out how exactly they dim the lamp.
It features AM reception, so I doubt it would use any type of a phase cur dimmer, as they would not be able to clean that noise out from the AM reception.
What remains is a diode (and that is questionable, as having such circuit so closse to the loop antenna, it would make additional parasitic modulation so a buzzing/humming noise) or a voltage reduction.
And that could be either by a tap on the transformer primary (a "10W" size transformer can easily handle dimming of a 40W bulb - at 70% voltage and below the transformer has to handle only about 20..25% of the lamp rated power). Such transformer supply would be the cleanest in the AM.
or some sprt of extra series impedance (capacitor?).
The diode could be a flicker problem for a standard LED designs.
The transformer would make the dimming working on a simple capacitive dropper style LED circuit.
The extra impedance will likely do nothing (no harm, nor any significant dimming) on all commercially viable LED ballasts.

Real danger for the LED or the radio internals would be the phase cut, but its use is very unlikely on a radio featuring AM.
So I may say you may safely try the LED, but only if you are not using, nor ever plan to use, the AM band. Because even just the rectifier will cause significant buzz/hum there.
FM (that includes the WB) is immune against this type of noise...

Why not just use a lower wattage incandescent so it doesn't get as hot?

Well I did try a LED bulb, and it does work on high setting. But I'm considering that too, with a 25 watt incandescent instead, which will give high and low setting still.

But what make me mad, is I went to take out the original bulb, to put in LED, and the glass is come loose from the base. So all that for nothing, as I ruined the original GE bulb.