At night fluid will accumulate in your spine hence the morning stiffness.
Without a proper assessment it is hard to tell if you have a disc protrusion. Perhaps do research on lordosis of the lower back and how to get rid of it i.e. pelvic tilt. I have excessive lower back lordosis which gives me similar symptoms and it was due to anterior pelvic tilt (sticking my bum out too far) as I had very tight quads from a jumping sporting background.
If you lie down flat and there is an arch in your back taking your lower back away from the surface you are lying on this is a good sign of excessive lordosis. Bringing your knees to your chest as you stated will flatten this arch and that what is leading my thinking. If this is the case perhaps sleeping with pillows under your legs to create this position throughout the night will ease some of the morning stiffness.
But as previously stated this might not be your problem at all and it is hard to tell without assessment.