I agree with Monica, basically. Changing the pedal slightly after the beat doesn't constitute syncopation--after all, if you tried to change exactly on the beat you'd get a gap. You have to change the pedal after the new chord has gone down so the transition is smooth. I hadn't considered playing this piece without pedal at all, and wouldn't perform it like that, but it is a very interesting idea for practice.
On the difference between the two editions: An 'Urtext' edition, as I understand it, is an attempt to reproduce, as faithfully as possible, what the composer actually wrote, without the editor making too many value judgements as to what he meant. (That's a bit problematic when you get to Chopin, who had a habit of revising his pieces after the fact, so there's often no single definitive version, but never mind that.) Henle editions are very highly regarded, also not cheap, but you get what you pay for. So you can pretty well take it that the Henle is Chopin's original pedal notation, and the others are editorial attempts to clarify what he meant by it.
Now your Romantic piano composer won't typically try to notate every detail of pedalling. As pianist, you're expected to use your discretion in the absence of pedal markings--it's not taken to mean "no pedal" rather "pedal ad lib." Not that most pianists don't tend to use too much pedal, but I digress. I should expect that, in the absence of any markings to the contrary, 99% of pianists faced with this particular piece would just change the pedal with every beat. Monica does. I do. So there's no reason to notate it unless you want the pianist to do something different. That, I think, is what the pedal marking on the last bar in the Henle is telling you. It's not an instruction not to use the pedal elsewhere; it's an instruction specifically not to change the pedal for the last chord, but to let the bass chord continue to ring as you play the last one.
(One of the stranger things I've heard lately is a Brazilian prog-metal band doing a rock song based around this prelude. It's surprisingly good.)