Hi. I'm a new user here and am just getting back into vinyl.
I picked up the very same model Sony PS-LX4 turntable earlier this month and it had the same issue with 45rpm.
Cleaning the speed selector switch was very straightforward - no service manual needed. With the unit turned over, you only need to remove the (6?) screws that secure the panel underneath the front, not the whole base panel.
One more screw loosens up the small circuit board with the switch and quartz lock light. Quick spritz of contact cleaner and exercise of the switch and it all worked perfectly when put back together, with the quartz lock light now illuminating on both speeds.
Hope this helps.
Next up for me is an elliptical stylus upgrade, on its way from Amazon, as the current one sounds dull and I have no idea how old it is. Mine has the original Sony-specific headshell with XL 200 cartridge and ND 200 stylus. I would have liked to try a new cartridge altogether but this would require a new headshell with a fitment that is smaller than standard. I want to see if the TT is reliable over time before I spend much money on it.