I hadn't realised this would be quite so tricky (and to be honest I don't think it is). I'm assuming that, for the majority of people, LEL would count as a July RRTY qualifier, so you would have to convince me that you rode at least 200km of LEL during August for it to count as an August RRTY qualifier instead - although I'll even accept "just" the 199km from Kirton-Loughton. That's at 12kph too, so it's quite a concession.
A bit of resourcefulness might be called for as you cannot (and must not) expect the LEL controllers to do anything other than their normal LEL duties. They've got to look after 749 others, and may well have had less sleep than you, so just don't push it.
Acceptable proof would be the timed stamp on your LEL brevet card at Pocklington, Market Rasen or Kirton, and your name on the LEL finishers list. (If you are in Thirsk in August then by my reckoning you are out of time)
Sadly, an abandon even at Great Easton is a great big DNF.
If you are really adamant that you need LEL for an August ride, it wouldn't be a bad idea to get something like an ATM slip or shop receipt from somewhere north of Kirton and dated August, just to verify your whereabouts.
For me, just finishing RRTY is a big achievement in itself, and worth far more than yet another RRTY. Anyway, I would hope to finish in such a condition that I would be able to ride another BR before the end of August.