Have you actually been assessed by a physio/sports rehab practitioner already? Who has told you that your right ilium is anteriorly tilted? Your SI joint relies on stabilisation for optimium transfer of loads ie hopping. This stabilisation comes from sorrounding muscles and ligaments If muscles around the pelvis/hip are not functioning properly ie 'tight' (over active), weak (under active) or lengthened this will lead to an structural imbalance around the pelvis and SI joint. This can lead to a functional leg length discrepancy. Anatomically your individual leg length's are probably similar. When you have muscular imbalance ie. shortened hip flexor musculature, this will pull your pelvis forward into anterior tilt. This in turn will affect the length of your leg in such movements as lunging (which brings on your pain) as your pelvis is unable to posteriorly rotate.
Your lumbo-pelvic ('core') stabilisers might also be weak. Your groin pain might arise from this. As the pelvis is unable to use it's 'core' muscles, it relies on the adductor muscles instead to stabilise during movement. The adductor muscles are not designed to stabilse the lumbo pelvic area as they do not attach to the lumbar spine, thus become over active and painful.
You need a full assessment on length of hip flexors, lateral rotaters ie piriformis (might be causing nerve pain down leg), adductors, abductors etc.
You also need strength testing on your lumbo pelvic stabilisers, glutes, adductors, hip flexors etc.
You need to be checked for functional leg length discrepancy.
As for exercises, you say when you purposely activate glutes, you can move pain free. You need to strengthen glutes (bridging etc) You also should look into Richard Jon Tigney exercises to teach your SI joint to move properly again. You probably need to lengthen right hip flexors (iliopsoas), piriformis and adductors.
Any particular reason or time when your pain come on? what mechanism?
SI joint pain is often insideous (gradual).
You will probably have to strengthen your core also. Maybe go back to basic's of activating your TA muscle while lying down and kneeling, learning to breath whilst activated. Work your way up to keeping it activated whilst moving functionaly.
Also get your lumbar spine properly checked out to rule out any reason for it to cause the pain.
There is an awful lot of if's and maybe's here as i cannot assess you myself.
Hope that helps a little.