Which LARS?
In your example defnitely Bristol, as Lyneham aren't even a LARS unit. They tend not to be that interested in you unless you are transitting the zone so you would be better off traffic wise talking to the LARS provider in the area.
If you were to look at the map available of LARS units in the AIP, Pooleys, Bottlang etc you'd realise that the overlap tends to be not much more than 10 miles between some adjoining LARS units, so more often than not it's pretty self evident who the best unit to talk to is.
Another thing to take account when you're planning is whether the service is available when you are flying. Filton for example is a LARS service 08-1800 Mon-Fri. Don't even bother trying to call for a radar service during the weekend however - the radar isn't even switched on.
It all boils down to planning - look at the books - see whose area of coverage you'll be in.
CM