A problem with key signatures

How about this workaround.

Both versions play the same correctly.

I copied the first stave to the second stave then on the second stave:
1. 3rd bar - select the 4 D naturals and increase chord x offset to make some room at the start of the bar.
2. 2nd bar - select the key signature before the repeat, turn off automatic placement, increase x offset to move it to start of bar 3.

I did this with Musescore 3.7

" I could find no workaround to this problem. I need the new key signature to appear (correctly) after the barlines"

There is a workaround:
1. Select the first note of the start repeat bar, in Appearance increase the Leading Space (e.g. to 2 or 3).
2. Choose the new key signature from the palette.
3. Select the new time signature, in Properties turn off Autoplace.
4. In Appearance, change the horizontal offset to move the key signature after the start repeat barline.

I usually support Gould's rules, but in the example above I think it's much clearer if the change of key signature comes immediately after the start repeat barline. That's because all the changes of key are shown within the repeat itself, so the player cannot miss the key change on the second time through.

This is certainly a special case that Mrs. Gould has overlooked. I don't know her work. But it's simply much clearer if you write it that way in this case. As important as rules are, clear notation is even more important for the musician.
I agree with both of you.