The fault is in the PT and probably unfixable Anakin's sword should have been an emblem of Luke's idealisation of his absent father.
It's no coincidence that the Vader delivers the bad news right after Luke loses the sword.
Luke discovers the truth about his father so his dream of a heroic father cut down by an evil other is as lost as the lightsaber is.
I like your idea of Vader taking up a new sword the moment he rejects his Jedi identity it makes a lot of sense.
One way around it could be that he doesn't actually have his sword with him.
He has left it behind when he rushes to the Mace/Palpatine show down.
Obi-Wan makes a big deal of him doing this sort of thing in AOTC.
So instead he uses one of the fallen Jedi's sabers to cut Mace's hand off and gets his new red one from the Emperor.
Swapping the saber grip during that swipe at Mace shouldn't be too hard (it's all work but compared to some of the other things you have done this will not be as hard...I hope).
Even if you can't get a line for Padme you might be able to find a shot of Padme looking off at the distance and becoming particularly distressed.
You could insert a shot of the forgotten saber in there not only is Padme worried that Anakin is in the burning Temple she has the added worry of him not being armed and fully able to defend himself.
Later Obi-Wan could pick it up in her room as he goes.
I wouldn't worry about the red saber having the same grip as his blue one as over time Vader may have personalised that saber or replaced it with the one we see in ANH (which isn't quite the same all the way through the OT anyway).