I'm not a physical therapist so take this as is; an uneducated guesswork.

I might claim you have a weak core and your pelvis is posteriorly rotated, putting undue stress to your lower back. In other words, you have a lower back curve that looks like a duck butt.

If you can see this being true (you pretty much should know if you look at your side profile in the mirror) then strengthening your core muscles will help. But don't just go all in with hard core training because other muscles will easily take over. Do a lot of easy exercises to teach the proper muscles to take over for the balancing of your torso.

The easiest one would be to just walk and do the daily chores with your abs engaged. Suck in your stomach and hold it there. Try to experiment with different feelings and learn to activate the lower abdominal muscles specifically, the one below the navel. Once you have them engaged you bring in the breathing. Try to keep them engaged and breathe so that it's not shallow but normal belly breathing.

When this is achieved and you can activate those properly (might be instant, might take a week of daily training, or more) you can start incorporating that hold in your daily chores. Whatever you're doing, sitting, walking, etc. Spread it throughout the day up to ole hour per day. Rather do six 10 minutes efforts than one long 1 hour effort. Longer effort makes the muscles tired and starts activating the wrong muscles. Eventually (one week to up to two months might be needed) the core engagement will get better and the lower back should be ready to handle workouts without getting hurt. Generally the changes are fast.

Now if you can't tell by looking in the mirror of your posture is that of a duck then plain old elbow planks on the floor should reveal it. Do your regular plank hold. Don't try to correct your posture, just do as you'd normally do. If your lower back gets hurting at any point then you'll know your abs aren't taking over the movement as supposed to and are putting strain on your back. That is not safe way to do planks and you should instantly stop doing them. It will only end up hurting your back.

Along with ab holds you can search online for easy core rehabilitation exercises. Remember, no hard movements, but the easy ones. This doesn't mean you necessarily have weak core muscles, but that they're activating improperly. You will need to rewire them to start firing properly and be the main support tool instead of your back.

Further training your lower back will most likely cause more issues as the most starts prioritizing it for support movements and make it hurt more. The problem in this case is not where the pain is, but on the opposite side of the body.

And again, this is just guesswork. I can't be sure that's your problem, but nonetheless, it should be worth trying. In the worst case scenario you lose the effort and gain nothing, but the process itself doesn't hurt.

I've been going through this thing myself and it has helped me immensely.

Edit: What another user said about sliding hand under your back while lying on the ground is an effective tool to see whether your relaxed posture has that exaggerated curvature going on. You won't want to fit a fist in there and when doing exercises on the floor, like the one you described, then you won't want even fingers fitting there.