Superlux - HD688

For 50€ you could get great IEMs that would work well for metal (too many options to list), you can even get TWS earbuds with noise canceling and good sound for under that price (Moondrop space travel). With over-ear it's harder to find something within budget. Best option I can think of right now would be something like the Superlux 688B + bigger aftermarket velour or PU earpads to boost the bass a bit and make them slightly darker sounding. They fit most generic round or slightly oval earpads. Take a look at muziker (you should be able to get both headphones and earpads there), or find the headphones somewhere else and look at aliexpress for earpads. You could also look at the used market for studio closed-backs. I got my K361 for 20€ (they came without a cable and I had to spend some time to make them stop smelling like vape). NTH 100 is good, but it might be hard to find under 80€.

Koss (75, 30i, 40) wouldn't be my first pick for metal and they're all on-ear not over-ear, but they are light enough that it shouldn't be a problem. They have basically no sound isolation though so maybe not ideal. I'd also recommend the 688B over the 681 having trialed both.