The original holes are too close together, so I will need to fill the holes with either wood filler or glue and dowel. Which filler I should use would depend on which would hold up being drilled to reposition the handle holes. Can anyone give me some advice, or what they would do?
either wood filler or glue and dowel
That's what I would do. I would choose a wood filler that says on the tin that it can be redrilled. You might be able to use a repair-washer on the back to spread the load if you are worried.
Otherwise I'd drill the hole out to the next largest dowel size, glue in a dowel and use a flush-cut saw.
If necessary, you can improvise a dowel by drilling a hole in any convenient bit of steel and using that as a "dowel plate" through which to hammer some slightly oversized wood rod. Chucking the wood in a hand-held drill can work to speed up the production of the dowel.