Naturally, the shop transitions would not count; they never used the "iris out" effect, that is a separate type of scene transition that cycles the color palette around in a strange way, and the bgm does not halt during such a transition (only once it's finished and the screen goes black). If you notice, the same behavior occurs when you go to the menu or enter the 15-number mini game on the boat. This "shop transition" and the "iris out transition," I must therefore conclude, are governed by separate routines in the game's program.
However, there are often "iris out transitions" where the music is supposed to change (typically when entering a location from the world map, or exiting a location to the world map).
I can think of two exceptions right away, though:
1. Mirage Tower to Sky Castle
2. Temple of Fiends (Present) to Temple of Fiends (Past).
Those are times when the "iris out" effect occurs, going from one interior location to another, and the music changes.
I could be wrong, but the simplest way to stop the music from restarting during an "iris out without track change" would be to just include an exception for every time it occurs in FFRC.
You wouldn't be able to automate the process any further on such old hardware, I don't think. Every time you're going from one interior location to another, you would have to direct the game to continue from where the music left off.
Notice, as well, that FF1 does not ever have such behavior in the vanilla version, at all. Whenever a new track begins playing, it is always started from the beginning. That means you'd have to code a stop fade instruction for it from scratch, and find a way to get the game to recognize it at appropriate times (when the player goes from one interior location to another without a track change).
However, the "No menu music" patch DOES produce this behavior, but it's during a "shop transition" (the rainbow colored palette cycle), the one that the shop doors and the menu share.
So, in conclusion, if you want to add stop fades to interior location transitions that originally used the "iris out" style of transitioning, I would begin with a message to Bregalad. Clearly, he is the only person who knows how to make FF1 do this, if not specifically in the situations I'm talking about.