Yes and No. I'm doing it on fixed. Plenty of people have done it on fixed in the past. It's not that tricky a fixed ride. But it is barking.
I did a fixed SR in 2008 including rides like the Dean (3900m climbing in 300km in the worst weather for 20 years of the event!) and the Severn Across 400 (similar to the Dean but not much more climbing). My 600 was a DIY ride Putney -> Cambridge -> Putney -> Royal Tunbridge Wells -> Bethersden -> Flattest Possible 300 (which isn't completely flat).
This year I'm doing, amongst others, a fixed Dean, Elenith (hello Mr Devil's Staircase), and Bryan Chapman Memorial 600.
The tricky bit of LEL will be the section from Canonbie to Dalkeith via Eskdalemuir. There's a fair amount of climbing on that section. Nothing amazingly steep but just long and constant.
My plan is to blast out to Coxwold (400km-ish) on day 1 (8am start). Day 2 is a punishing day over Yad Moss, Alston, Brampton, Longtown, Canonbie, Eskdalemuir, Innerleithen and up to Dalkeith. At that point I'll have lots of time in hand and I can have a gentle 200km day 3 back over the hills. Then another blast along the flatlands for 300km with a gentle final day.
In reality I'm going to ride until I'm tired and then rest until I'm feeling better, or I'm on the edge of the time limit, then start riding again. This is the best plan to have anyway.
Do it. You know you want to.