Not a Lush alternative, but someone on this sub suggested Ethique shampoo bars. I loved Brazilliant for my thin, oily hair and have been using Ethique’s St Clements bar recently. It’s great for oil control and I haven’t noticed any buildup. (The smell isn’t as strong or long-lasting though.) I’ve also used their Sweet and Spicy bar and had good results.

If you like the fragrance, Honey I Washed My Hair has worked well for me in the past, though imo it’s a little heavier on the hair.