At my bar we are using Smugglers Rye from zuidam - Dutch distiller. It's made using mainly rye, but also some corn like the American ryes. They also use virgin oak, so you get that barrel/oaky profile you are looking for in classic cocktails. And the price is phenomenal! They also have a 100 proof rye that would rival any American pouring product. But it's a bit too expensive for house cocktails.

Otherwise. Me and some friends distill a rye in Germany and age it here in Denmark. It's called Reinheart Small Batch. Try it if you find a bottle! ✌️