Biking with spinal disc herniation
To be very to-the-point, in your situation I would first consider the following two things:


Rise/exchange your handlebar to leave it at least some 8 inches above the saddle height;
Have a good quality suspension seatpost.


I have studied medicine (and have a little back pain, also), so the rationale is the following:



This figure illustrates a part of the lumbar spine, as would be seen if the rider was passing from right to left on the bike (left is front, right is back).


Rising the handlebar would allow you to be more erect, thus making the upper vertebrae rotate backwards (clockwise in the figure) over the lower vertebrae. Since this rotation is articulated a bit behind the midline, there is a decompression of the discs (blue and red, with red being herniae and blue being normal disc tissue);
Since your bar is now higher, a greater part of your body weight is supported by your back/buttocks. Using a suspended seatpost - a relatively lowcost, minimal modification on your current rigid bike - helps avoding unpredicted/unavoidable impact, which is a main causation factor of hernia inflamation and/or progression, due to the sudden rises on internal disc pressure associated with impact.


I don't enjoy very much the feeling of sprung saddles (I have a Brooks Flyer), and I think that's because they pivot around the nose, thus bending the lower back and pelvis while going down, in a flexing motion. Seatposts, on the other side, have a translation movement, without pelvis rotation.

Having ridden some suspended-seatpost-equipped bikes, I can say one thing: they are awesome! And since you have a condition, thus are probably not wanting to become world champion anymore, the minimal, questionable performance decrease is overwhelmed by the very significant gains in comfort and health risk reduction.

Hope this helps!