You could train around all that with stuff like belt squats, box squats, good mornings, high rep glute bridges, kettlebell swings with a band around the waist, kneeling squats with a band around the waist, etc. With the belt squats you'd want to mimic your sumo deadlift stance, same with the box squats, above parallel box a little below where your deadlift starting position would be. Building glute strength might help with the hip pain, too. I still think less complex programming is your best bet for now, but you could temporarily sub in some of these movements for deadlift for awhile till you can train normally again. Might even progress your deadlift without actually doing it.
Like for example for the DUP setup instead of deadlift you might do box squats on power day, good mornings on strength day, and belt squats on hypertrophy day. Do the direct glute work on your accessory day.
If your gym doesn't have a belt squat machine you can do something like this with a calf raise machine and dip belt. It's a good way to strengthen your sumo deadlift without getting your lower back fatigued.
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For the kettlebell swings you'd just tie a band to something behind you and put the other end right around your hips so when you're swinging the kettlebell bell out you're driving your hips and glutes through against the band resistance. It's worth a try along with the bridges and mobility work.
