The seat panel is very unlikely to have lateral forces exerted on it during use. The only significant forces are likely to be up and down.
As per your second pic, https://ibb.co/bBttFd5
, the seat appears to be hinged at the back, denoted by the 1, 2, 3 "markers"/tags, but I see no hinges. Is the seat attached, in some way, at those points?
If there is some sort of permanent attachments at those 3 points, then you only need one other attachment, if that. A seat does not need to be secured all the way around the frame circumference. It is unlikely the seat will move, no matter what a sitter will do. A seat as that only needs a minimal lateral security. If the back edge is hinged or "attached" in a fixed position, no other fastener is needed for the seat panel to be stable.
I'm not sure why the whole seat perimeter was glued down to the whole seat frame. The pics don't give an indication as to why, so I may not see or understand why it was glued all the way around. I agree with you, that the whole perimeter gluing attributed to the seat cracking.
If the seat panel is secured at the back edge, in some sort of hinged fashion, then the rest of the seat's edges don't need any attachments. The seat panel is not going to move laterally, nor will the panel likely crack again.
These above assumptions/suggestions may be different if further pics show some other seat-to-back edge attachment or arrangement.
Another possible option is to glue two small thin blocks on the underside of the seat panel at/near and just within the front edge of the frame, such that the blocks prevent lateral movement of the seat panel, assuming the seat panel is hinged/affixed at the back edge.
One other note: I googled Harvey Nichols chairs and got no specific hits relative to your concerns or its design. That particular chair's design may have been so faulty or had proven to be so problematic, that the chair/design was abandoned/discontinued. This may relegate it to be rare and worth restoring or preserving, as best you can, in its most original state. More pics may help resolve your repair/restore problem, rather than just "lets get it in seating condition".