"British democracy" is a rather strange thing in itself - and if you look at how our system works (and has done for many, many years now), it's already well and truly broken. In the confines of our current system though, we elect representitives to make the tough decisions for us, in the best interests of our country, specifically to avoid situations like this when the general public are well and truly split with no clear majority for a decision either way (you can argue the 2-3% thing as much as you like, but practically the country is split and leave or remain, people will be pissed). That system already isn't "democracy" as many would understand or like it, and thats before you even get into the mess of our electoral system that supports it.
The very fact the referrendum was positioned in the way it was has been really damaging and has left the government tied in knots, as they know Brexit is a huge risk and that it is likely to be initially hugely damaging to the UK, but they can't not try and follow through with the result as-is given the promises that were made, even if it's actually their job to make the correct & best decision for the country, which historically would be to remain given the general propensity to avoid risk in large government decisons in the past. (no one knows for sure if leaving the EU will be a good or bad thing, whichever way you voted - anyone who says they do is making it up).
I am clearly a remainer and have never heard of a logical, rational and non-angry argument for leaving the EU, but I also don't want a second referrendum. Personally I want the governement to do their job and act in the best interests of our country, which is what we employ them for, rather than all this pandering to a rather pointless vote. That would be to abandon this and remain....With a little luck and 2-3 weeks of policy making this whole disaster could be undone. Sadly the majority of our MPs are too spineless to do it, thinking of their own careers rather than their jobs (a strong concept in many ways
) and i'm left unsure of what will happen this week.
tl;dr - I believe that the truly "British Democratic" thing to do is to ignore the referrendum and remain tihe EU. Thats not the same as the by-the-book "democractic" thing to do, which would be to leave based on a slim margin of a public vote