I have a bulged lower disc above my tailbone. If I go a week without exercise the muscles (butt & lower back) start to get tight and put pressure on the disc, thus on the sciatic nerve. The doctor told me I will have to live with it for the rest of my life. I find that exercise is the best medicine. I have been following Mark's PBF 2x a week without any sciatica except when I traveled to Asia where exercise was hard to come by. After just one day back in the gym my back pain was almost gone.
Someone else said CrossFit - no joke. I agree - any type of exercise will keep your muscles loose - not just cross fit though.
buy a firm foam roller and roll out your glutes, hamstrings, back, etc. Google some videos to get the idea. Almost as good as a having a sports massage. Helps heal via bloodflow and breaks down scar tissue/fascia. The more uncomfortable it is, the more you need it.
I can't find major fault with your workout -the volume's not too high and the balance push to pull seems OK.
One thing to remember, we need to build in deload weeks. One full week every 4-6 weeks where we don't lift at all, or do a similar routine at 50%. If you haven't deloaded in a while, you might need a longer one.
I'd give your back a month to heal. Make sure you're getting healing sleep. Keep up the ice.
Right about now (if you were my client), I'd start you out with bird dogs, single leg deadifts, cossack squats, and step ups 3x a week. Body weight. Still sets of 3x5. I'd change your chins to pull ups to work more on engaging your latts. Your workout seems completely saggital (only in the front/back plane) so I'd discuss with you ways to add side-to-side and rotational movements to your lifting. (it helps prevent injuries).