all i'm asking for is one way to justify to the authorities that it was only practical for me to leave 1 day late. i'd much rather do this early next week with the Home Office/IND than to have to do it at the airport next sunday. any ideas with this?
You can go to Lunar House with an FLR(O) form and explain it to them, or you can go to the IS at a port. They will be violating HC395 if they extend your stay beyond six months simply because it was inconvenient for you to book an earlier flight - which is what it adds up to in their eyes. But given that it's a single day, there may be a finesse play on the day-counting - but that's the sort of thing an advisor should do - i.e., someone with credibility on the line. Otherwise they won't do it.
They will also look at your frequency of travel in and out of the UK, which you didn't tell us. And what you were actually doing here is also missing. It's natural for them to assume that you were working illegally and you would expect a question or two along those lines. So [1] if you have lots of prior visits in to the UK; and [2] you plan on visiting the UK in the future, then catch an earlier flight.
If you are not planning on coming back, then it might be more expedient and cheaper to sort it when you depart.
Interestingly, I don't have any experience with single person, single day overstayers, mine are months and years, and include whole families :\\\'( . For that reason, it would be helpful if you kept notes along the way and got back in touch after your trip. Use PM if you like!
To answer your other question, yes, it's a big deal.
Editted to add: my wife wants to add that you can also exit to the EU on time, catch a flight that connects to your original flight and use an airside transfer at Heathrow. All in it costs about £50 round-trip. Shrewd she is.