Thatcher was not 'well on the way to dumping FI'.
There was a belief that the islands could not be defended properly without the expenditure of money we didn't have and a notion that they were more bother than they were worth. This led to explorations of 'leaseback' - which were torn to shreds by Tory backbenchers when Nick Ridley raised the idea. Thatcher, at the time, had a majority of 43 and a wing of the party which was deeply uncertain about her leadership.
She also faced a Labour party led by Jim Callaghan who would have led a fairly significant number of his MPs into the lobbies against any notion of sovereignty transfer followed by the UK leasing the islands back, and a number of Liberal (as they were at the time) MPs such as Russell Johnson were against the idea.
The FIGC, ironically, was not unanimously against the idea, but the opposition in parliament to the idea meant that unless and until the islanders were in agreement, there was no chance of the islands being handed over. This is all online as well; sovereignty transfer and lease back needs to be placed in the wider context - which was that the Thatcher government was nowhere near 'dumping the FI' - not least since MT was busy minuting proposals from Carrington with lines such as ' I could not possibly agree to the course of action the Foreign Secretary is proposing' and 'NO. It would never get through the H[ouse] of C[ommons], and rightly so.'
[See National Archives, PREM19/656 folio 118, 20 Sep 79]
Then in Feb 82, Lord Carrington sent her a note -
http://fc95d419f4478b3b6e5f-3f71d0fe...2C6C87933F.pdf
Which rather suggests that the problem was nowhere near a diplomatic resolution; a memo went in the opposite direction reinforcing the view of the PM that the wishes of the islanders remained paramout.
Other documents, in Kew, but not in the Thatcher archive also suggest that her administration was nowhere near 'well on the way to dumping the FI'.
Like the proliferation of comment about the independence of the deterrent, a lot of the material out there is well out of context; unlike the Falklands, the actual documents demonstrating that the scenario is rather more complex than portrayed are not in the public domain...