Frankly, what you have been told is a white lie. I have heard that line from other airlines in the past. Airlines want to discourage people from leaving the airport while in transit for fear that the passenger does not return in time for departure.

The reality is that VS has no means to check whether you stay at the airport or not and will not offload your bags simply because you leave the airport. So, just check your baggage to your final destination. Do not bother discussing with the VS agent whether or not you will leave the airport. On arrival in London, do what you want to do during the day but do make sure that you return in plenty of time for your return flight in the evening. This is the only thing that really matters.