Battle.net opens at the focused monitor's scale
On a HiDPI monitor the Battle.net launcher opens at 100% and looks half-size. A change merged into quattro and into v4-5-0 sets the DPI from the focused monitor's scale before each launch.
Hyprland leaves these XWayland apps unscaled, and Wine scales them from a DPI value the prefix keeps at 96. ryanrhughes merged the change. Scale times 96, so 2 becomes 192 and 1.6 becomes 154.
He tested it on a 4K monitor at scale 2. The launcher was half-size before, and after the change the registry read 192 and the launcher opened at the right size from the menu.
Wine reads that DPI only at startup, so a scale change waits until the next launch. The value is left alone while the prefix's wineserver is running, because the running launcher would ignore it and the wineserver would write its copy back on the way out.