The SI joint is a joint that wants a lot of stability and very little movement. Some of the classic "core muscles" help stabilize it like the multifidus and transverse abdominis help to stabilize it and there are a lot of ligaments associated with it. Static core movements makes sense to reinforce this stabilization so that is why you physio recommended them. Also the dead bug track would not be out of place. Wait for you physio to tell you the next step but I would like to mention that static stability is only part of the package. This should graduate to creating active stability.
You should be able to keep doing lighter squats and deadlifts. I would skip the bent over rows like you said. You can do all sorts of other stabilized rows and bent over rows aren't that important in the big scheme. Mobility drills would have nothing to do with the SI itself but creating the proper mobility in the joints that should be mobile, such as the hips and ankles. Standard lower body mobility drills that address hips and ankles are fine.
Check for hip flexor tightness. You may need to specifically do some hip flexor stretches as tight hip flexors (iliopsoas) may be a factor in SIJ problems. Alternatively, tight hamstrings could be associated with it so look for that an if needed incorporate hamstring or hip flexor stretches after your workouts.
And specifically for squats and for added mobility look at the advice I give in "Tweaking the Overhead" squat.
SI joint pain may be made worse by sleeping on your back. Sleeping on your sides with a pillow between you legs and people who have low back pain and find that that method helps may be experience at least some of that pain from SI joint dysfunction and this may have been misdiagnosed as some sort of lumbar nerve compression.
It actually takes a whole lot of different diagnostic tests to tease out SI joint dysfunction from other problems that cause similar symptoms, so I hope your physio is minding their P's and Q's