In my experience, the "Basic Service" from a LARS is often much more than that, around London at least. Twice in the last few months I have been overtaken by a faster aircraft on the same track. Three-way comms made it much less stressful.
The first time, LARS had little else to do, and conditions were good, it would probably have turned out fine anyway.
The second time (in marginal VMC) LARS was very busy but we pilots figured out from our own calls what was happening and rapidly got some helpful attention, despite the "workload". Basically just a "G-XX, G-YY will pass on your right" was enough.
It isn't guaranteed (I have met an opposite track formation without a peep from ATC), but why pass up any chance of timely assistance?
I would always call.