Does prayer inside the Hijr while circumambulating from outside it (entering from the side not considered part of the Kaaba) interrupt the tawaf, and is it permissible?

The "Hijr" (Hateem) is part of the Kaaba, so circumambulation (Tawaf) from inside it is not valid. Continuity (muwalat) is a condition for Tawaf according to the Maliki and Hanbali schools, and a slight interruption is excused, unless a fard (obligatory) prayer is called or a funeral procession arrives, in which case one prays and then completes the Tawaf. As for the obligatory Tawaf, it is not interrupted except for a fard prayer or a funeral prayer.

The Shafi'is consider it disliked to interrupt Tawaf without an excuse, out of consideration for the differing opinion regarding the obligation of continuity. If a fard prayer is called, the preponderant opinion is that one interrupts Tawaf with the intention of returning to it after the prayer, resuming from where they stopped. As for the funeral prayer, Tawaf is interrupted for it because it is short. It has been reported from the Salaf (early generations) that a slight interruption for rest is permissible.