Nope, sorry.
Figure 90 minutes to 2 hours in many cases to clear getting to the terminal (arrival time is touchdown, not after all the taxiing and any busing), the long long walk to immigration, the queues at immigration, immigration and clearing the Customs lane into the terminal.
An hour each way on the Tube into somewhere in central London, and 2 hour check in for the flight across to Paris, and you are toast even though the Piccadilly Line goes every few minutes.
That's 5:30 in best case and you haven't found anywhere yet to raise that pint.
You could, I suppose, try the Heathrow Express, which takes 15 minutes into Paddington Station on the outskirts of central London, and 15 minutes back. It goes every 15 minutes so if you just miss them (you are so short of time you need to plan for disruption and just missing one or both.
So now we can squeeze an hour out, and you are looking at using 4:30 of your 5:00. I'm sure that a bar at Paddington could serve you a pint.
Is that the sort of stress you want? How much of a risk taker are you? It will also depend on which terminal at Heathrow you arrive at, and the terminal you depart from. Is it the same one?
I do know that there are literally dozens of famous stores at Heathrow and many watering holes which are happy to serve up a pint.