Could you replace the socket in your lamp with something like this:

https://www.amazon.com/GE-3-Way-Lamp-Socket-54372/dp/B002DN4LX4/ref=sr_1_2?dchild=1&keywords=lamp+socket+and+lamp+parts&qid=1635766701&qsid=137-4550665-3344823&sr=8-2&sres=B002DN4LX4%2CB01M1LNGFO%2CB000LNOISG%2CB008DB3G2C%2CB088LWXYNJ%2CB07CZPDFMR%2CB07LBVYGWG%2CB002DNIBEY%2CB01LY0UN51%2CB01E9HS6LE%2CB085QD6KT8%2CB002YENA0M%2CB01M0ZKM3H%2CB01D1DO89Q%2CB07HFPLJXG%2CB008UW8FA2%2CB07G4T47F5%2CB083587T29%2CB000H5QVEW%2CB003725RC4

(This link was easier than to post the image; you can probably get it at your local hardware store).

Sure, change out the bulb socket with a 3-way socket, and replace the power cord with new (or just leave the cord switch on).

I'd replace the cord switch with a dimmer (example) and continue to use a standard, dimmable LED bulb instead of a 3-way.