I love powerlifting almost as much as cycling. If you want a safe and fast recovery, try not to hit your max PR, but stay maybe at 80%. I find if I don't push too hard I'm able to recover a bit faster so I can go cycling the next day. Squats for me take at least a day of recovery. So I try doing squats at the end of my weekly workout routine and then take a day off the next day. Also if you try doing a max PR day, just reduce your rep range. Lastly do lots of mobility stretches and use a back roller. I've been using one for years and my back is rock solid. I clock in at around 150 lbs. and I can easily deadlift 400 lbs., almost hitting 500 lbs. if I try PR'ing.
On a side note, I'm a big believer in strength training for cyclists. I know way too many cyclists who've had minor crashes and end up going to the ER with severe injuries and broken bones. Back in January I had a bad downhill crash going almost 30mph. I hit the pavement hard, gashed my thigh and elbow. I was a bloody mess, but I didn't break any bones. I think lean muscle mass and higher bone density has been very beneficial in reducing injury and speeding up recovery.