I don't do yoga ordinarily, but I'm prone to similar pain due to a compressed nerve that flares up in my lower back. I have a set of yoga stretches that relieve the pain pretty quickly.
I don't have a chronic injury or anything like that. Just routine sciatica that flares up occasionally. But it's startling how quickly the pain will resolve if I do the stretches carefully and regularly throughout the day. If you do a Google Image search "yoga for lower back pain" you'll find a lot of pages suggesting simple and safe stretches. And youtube has lots of help as well. I carefully avoided anything that looked challenging because I am not a bendy person at all.
I tried massage but it didn't help. Except that the massage person recommended that I just slow down. And she's right. For example, getting clothes out of the bottom of the washer. Just slow down. Reach in carefully, don't twist, maybe even get a little stepstool to reduce the bending.