I pretty sure I saw Phantom F.3 used officially but it was ignored in practice because F-4J(UK) was already in wide use. With the US and UK Rivet Joint aircraft being so intertwined I suspect what ever name the Project Airseeker aircraft were intended to have was quietly forgotten.
Presumably their Airships have a cultural aversion to numbering reused names.
There seems to have been a run on the names of Hawker's Hurricane replacements none of which were II, III etc.
Hawker Tornado - prototype flew but cancelled because RR Vulture scrapped because of the problems in Manchesters (actually worked fine in Tornado)
Hawker Typhoon - the original Tiffy - parallel development to Tornado
Hawker Tempest - ironically briefly Typhoon II
Reused as:
Panavia Tornado
Eurofighter Typhoon
XXXX Tempest
Their Lordships used to have a tendency to have US aircraft with different names e.g Grumman Martlet = Wildcat.
So to avoid confusion over L or L II or F-35B (or even Dave) I suggest in future the UK uses the name of the final descendant of the Hawker Typhoon ...
Lockheed Martin Sea Fury FGR.1