I finally discovered a 3 way LED with mogul base, made by Satco. I discovered these last week on Overstock and Wayfair and now on this site.
They come in 4 different color temperature options. They are a little ugly but in a Torchiere or center bowl of a 6 way type lamp you wouldn’t notice it I don’t think.
It’s also interesting these use brass colored base threads like some incandescent mogul bulbs do.
A little pricier than a typical medium base 3 way LED but a good option to be able to use the 3 way mogul socket to its full potential while updating to a more energy cost efficient bulb. I’m not sure if these just came out or not but I know I’ve searched several times and never came across them before.
https://gellerlighting.com/search.php?search_query=PS25%20satco§ion=product