How to hear system audio on JBL headset while also using an Apple Watch

Question in short:

When both my apple watch and JBL bluetooth headset are connected to my iPhone, I only hear music through the headset, but notifications only play sound on my watch. I want to hear them in my ears.

I'm using:

• iPhone8, iOS 15.1
• Apple Watch SE A2352, iOS 8.1
• JBL 125tws bluetooth headset

When I'm using my headset, it plays music fine. However, I don't hear the sounds of notifications. Those are still playing on my watch. Because the headset is in-ear, I don't hear the sound. I get a vibration on my wrist, but I sometimes miss it. Most messages aren't urgent, but recently I purchased a Ring doorbell and I already missed someone at the door, because I didn't notice the notification.

I've tried using the headset with the watch turned off and then I'm hearing notifications in my ear. So it's clearly a preference of the notifications to be played on the watch instead of the headphones. I want to change that preference to the headset when turned on. Is there any way to do so?

I've searched lots of settings and searched the web for support, but couldn't find anything.

Btw, when logging into this forum I got a 2FA-number, but that sound actually played on both my headset AND my watch.