...listening at a distance of ~2.7 m with small 3" full range speakers and a sub with high pass and low pass filters...
I appreciate the metric system you use in Germany. The Imperial system is more confusing. Anyway, the Dayton Audio RS100-4 is a 4" driver, not a tiny 3" that grabbed my initial attention....Dayton Audio RS100-4. They go easily up to 20k. Not as smooth as a dome tweeter. Higher directivity, too. But they measure and sound quite excellent (if they don't have to perform bass duties)...
Adding a tweeter and crossover probably wouldn't kill your budget, but I appreciate the appeal of low budget. Although I too use a Wiim streamer, I can't bear to part with my Chromecast Audio streamer.
Sorry I continue to veer off topic. My A5se is too perfect for my needs and low budget to even discuss. I'm just a happy customer.