You should be able to use it in either A(Auto) or M(Manual) modes. These are the settings on the flash, not the camera. The TTL feature won't work with the LX3.
You'll need to use only the modes on the LX3 where you can control the aperture, meaning Aperture Priority or Manual exposure modes. You have to be sure you set the ISO settings on the camera and flash the same, as well as setting the f-stop being used on the flash to match the aperture set on the camera, or your exposures will be off.
If you know how non-TTL auto or fully manual flash exposure works, the SB15 should work, but the flash tube on the 25-30 year old SB15 today probably will not put out it's original rated power, so it'd probably be a good idea to test it with a flash meter to see what the true guide number for your unit is today. I have no idea as to the voltage put out by the SB15, but it's fair to say it won't pack the same punch an early Vivitar 283 would. There was a time in my life when I used an SB17, which was the F3 version of the SB15. Cute little flash with a neat bounce head, but if you bounce the flash, your working distance will be severely curtailed.