So I'm currently reading the Instrument of Government (as you do), and in articles IX and X (that deal with the composition of Parliament) is written:
the persons to be chosen within England, Wales, the Isles of Jersey, Guernsey, and the town of Berwick-upon-Tweed, to sit and serve in Parliament
and
That the persons to be elected to sit in Parliament from time to time, for the several counties of England, Wales, the Isles of Jersey and Guernsey, and the town of Berwick-upon-Tweed
Was the town of Berwick not part of England? Did it have a special status similar to the one of Jersey or Guernsey? Because when the Instrument lists the constituencies of Parliament, it's not singled out , it's in the middle of the list between Newcastle-upon-Tyne and Oxfordshire.
At the time it was disputed whether it was De Jure part of Scotland or England+Wales, due to conflicts in the previous centuries. It was a county in it's own right, so legally distinct from neighbouring Northumberland. There were many other "county corporates" at the time, but every example I know of has been given that status to help empower the town to deal with an issue or to recognize its prominence. Therefore, I think Berwick is unique in that it was given this status as a result of the border dispute, and was specifically named in this instrument as a result, re-affirming the English claim to it. The matter wasn't fully resolved until after the Acts of Union, as it happens.
Unfortunately, Hansard's online archive only goes back to 1803 and I doubt I'd be allowed access to any records that old that still exist even if I was in the UK, so that's about as much help as I can be.
edit: Anyone at Oxford should have online access to some records from this period, they have a fantastic library.