I have a table lamp that switches to dim-medium-bright-off just by touching the base. You don't need a 3-way bulb; an ordinary incandescent 40-watter will do. I've bought a couple of equivalent LED bulbs, hoping that they would perform similarly, but they don't. You only get bright or brighter, not dim.
If there's an LED bulb that works in this type of lamp, I'd be thrilled to know about it.
Did the LED you try state specifically that they could be used with a dimmer?
Some LED's are not compatible.
The touch lamp is essentially a three stage dimmer, so you'd need a dimmable LED.