Personally, I'm not a big fan of changing around canon music too much. To try weighing in specifically on things:
Ruins of Alph absolutely shouldn't be changed, since the inside of the ruins have a unique variant of the theme used outside.
I would also consider keeping Union Cave the same, since it's the first mandatory area that has that theme. Slowpoke Well is right next to Union Cave so if that's kept the same I would also keep Slowpoke Well the same for consistency.
Goldenrod Underground is so inconsequential that I would honestly consider it weird to give it new music.
Mt. Mortar and/or Whirl Islands could both use different music though, I don't really think of "oh yeah, the Union Cave theme is playing here" when I think of them. Burned Tower has unique music in GSC, doesn't it? I could see that working in Whirl Islands, and most players probably wouldn't even think "oh, they changed this music from the original." I could see a similar thing happening by changing the Mt. Mortar theme to Mt. Silver's theme.
Ilex Forest I would try not to change, and Tohjo Falls is also really inconsequential so I don't have an opinion on it. Perhaps change Tohjo Falls' theme to the Kanto Cave theme, since it's meant to be the first barrier you cross to reach Kanto.
By changing Mt. Mortar, Whirl Islands, and Tohjo Falls, you've now reduced the number of areas Union Cave's theme plays down to five without adding any extra music to the game that would stand out.