NeverarGreat said:
What if they switched lightsabers at some point? There’s actually a few shots during the table fight where Obi-wan is holding Anakin’s lightsaber, but then there’s a shot in between where he’s holding his own so right now it’s a continuity goof.

Holy crap, how does anyone even catch that??? You just marked yourself as a major geek. 😉
(Nothing wrong with that!)

You could re-edit that section of the fight so that they lose their lightsabers, and show Obi-wan Force-calling Anakin’s saber to him.

Not a bad idea!

Or . . . what if I have Obi-Wan going back to Mustafar later on, and finding Anakin’s saber along the shoreline? That would not only explain how he got the saber, but also how he finds out “Darth Vader” is alive. Even the original ROTS implied that Obi-Wan assumed Anakin had died on Mustafar, so I guess we’re left to assume that as reports of the emperor’s righthand man “Darth Vader” began floating throughout the galaxy, Obi-Wan eventually heard about it.