As stated in the title, with the release of the Big Music Update, is there any way to play the music that would play during a loud heist while still in stealth rather than the stealth music?
Not within the game, no. It will only start the music when things go loud. However, you can use the Steam Music Player1 from the in-game overlay to play that music, if you haven't moved it out of the default location within the game's installation folder.
1may or may not require you to join the official Steam Music group