Yes, the early missile was apparently considered for the Sea Dart role. I have only seen this mentioned once, however, IIRC in Butler.
It is not clear how this would have worked, given the disparity in range. Guidance would not be an issue, the same system could be cued by radar as it was in Darkfire, that part at least is very simple. But Sea Dart was intended to improve performance against faster strike aircraft and newer anti-ship missiles, especially in combination, so long range would seem to be a requirement if the goal is to prevent the missile launch.
No the answer goes something like this.
PT.428 was considered as was a scaled up missile using SARH based on the aerodynamics.
But PT.428 fell to Mauler and Sea Mauler. Several studies included the latter during the effort to design supplemental ships as Type 82 spiralled out to an 6,000ton expensive ship.
Building on PT.428 work PX.430 was developed. This would become Sea Wolf, the ultimate expression of work going back to Popsy and heavily defined during Orange Nell, as an anti-missile-missile.
Moving from Beam Riding to ACLOS. Though I suspect a lot of the computer side was reused, were possible.
Meanwhile Rapier was emerging as an Army system. With features permitting a near PT.428 level of Capability if funds were forthcoming. At first they weren't.
In the Tripartite studies 1969, Rapier was dismissed 'as is' since the Navy felt it needed a bigger warhead.