The ideal way is indeed to have rips of the exact data from the cart, and possibly an emulation of the music setup if that changes how things play out. There are many ways to get a copy of an audio track from a game without that though. In this case HDMI, being a digital signal, should in theory be the exact sound the devs wanted when the boss came knocking and said "we need to have this out of here by [this day], get it done". More historically a line out cable* would have made something more than workable too; the devs would have hopefully used the old audio engineer's adage of make it sound good on crap speakers as that is what most of your listeners will have (and they knew enough to take advantage of the crappiness of TVs of the time --
http://bogost.com/games/a_television_simulator/ ).
Like was mentioned in the previous post there are also ways to manipulate the hardware outside of fiddling with the code running on it. For instance the megadrive/genesis has two audio devices that go to make the sound and if you find that one chip or the other deals solely in what you want you can disable the other (desolder the power leg or something) and rip away without knowing the first thing about ROM hacking. The 3ds had the headphone volume mod, something which likely happened again here and could prove useful for certain things. If the switch ever gains bluetooth audio support then that could also be something. Any number of devices have had the cheap digital to analogue converter (usually called a DAC) replaced with a better one, any number of devices have had additional electrical noise suppression figured out to help with this sort of thing.
*a popular way of doing things, too often though I see people that don't really know what they are doing make bad rips and share them. To that end the OP might get a good setup and try that, and presumably if the sounds were good enough to listen to in the game they will be good enough to listen to outside it.