I found a solution, albeit it might not be ideal for some.
Instead of using the alarm app, use the reminders app, as the reminder app plays only through your headphones while also letting you set a time for the alarm.
The only 2 downsides that I can see could cause issues are that it plays the set sound bite once, instead of looping it like the alarm app (solution is probably put multiple reminders back to back), as well as that the reminder app splits time into 5 min intervals, preventing you from say, setting an alarm at 6:48 for example.