Hi,
My initial response would be to emphasise stability exercises and leave theSIJ alone for a little while until you have corrected the strength/control issues. I agree that you should not be doing leg lifts....they will exacerbate the overactive Q?L and psoas. I would consider giving you a strong trocanteric belt to stabilise both SIJs while you do your exercise programme and I would probably be OK with releases but not stretches to the Q?L and psoas
Also your physio should check for any factors that put an assymetrical load on your pelvis This can be internal factors such as slight leg length differences, one foot pronated more than another. Also there can be ergonomic factors such as your work station/computer setup not being ideal especially if you need to sit a lot and twist to one side for work.
Another thought would be for your physio to show you how to reduce your SIJ yourself it can be very effective. This should be done very carefullyunder your physios supervision and only once you have more neuromuscular stability.
good luck! I have seen many of these types of back problems resolve with good management and dedication to an appropriately prescribed exercise programme.