"الحاج" in the given phrase functions as a gerund اسم فاعل, while "الحرمين" as an accusative. it may well be rephrased to: من يحج إلى الحرمين or, more tersely, to: من يحج الحرمين. other examples could be: الغاض الطرف, القاصد الخير, الكاظم الغيظ, etc.

the established english structure of object + verb + (er/ing), as in "baby-sitter", "moonwalker", "chimney sweeper" or "heartwarming", with or without the hyphen, is the best equivalent. an alternative could be--as suggested by other peers: nominative + of + accusative. but the problem here is with the proposition; a pilgrimage should be TO not OF someplace, hence: "the pilgrim to the two sanctuaries", which is a bit bizarre.

lexically speaking, sanctuaries could be shrines. the saudi king is interchangeably titled: the custodian of the two sanctuaries, the custodian of the two shrines, the servant of the two sanctuaries, and the servant of the two shrines. also, as a culture-specific term, "الحاج" could be just romanized as done above, and i recommend double a, to render the vowels more accurately, and to deffrentiate it from the noun "الحج", the pilgrimage itself.