I can't help thinking that the Absolute Minima thing is a red herring and simply confusing the issue. Absolute minima are an ATC thing and have no bearing on how
a pilot operates an aircraft. The absolute minimum is supposed to be the lowest minimum that any pilot can ever calculate, therefore it may be lower than the minimum in a particular Ops Manual or AIP plate. The idea is that if a pilot has calculated a minimum below the absolute minimum then he or she has made a mistake and ATC pass the message as a prompt to encourage the pilot to check that the minimum has been correctly calculated. The absolute minima procedure/message is not followed if Cat II/III approaches are available - usually indicated by LVPs being declared in force.
As to the original question, to me as a simple controller, it's all very straightforward. If the visibility at the time the approach is being made is at or above the minimum set out in the pilot's Ops Manual for a Cat I approach then the approach can be made. It really doesn't matter what's going on on the ground - LVPs or whatever. OK, I know there are some variables like whether the flight is operating to JAR-OPS and there's now this thing about the OM or equivalent position but as a controller I don't worry about them. Then there's some legislation about the approach ban that applies to all aircraft in the UK and UK registered aircraft wherever they are just to make the situation even more complex.
All this is only valid for the UK of course. And I don't think any other country has introduced anything as daft as the absolute minima procedure for ATC!