I would most likely check the following in the (advanced) adapter settings:
- no annoying power saving settings activated?
- U-APSD support should be deactivated
- Transmission power set to highest value?
- Transmission mode set to 802 11a/b/g?
Thank you! Tip from Asus support was successful: firmware update (important: download firmware from support website, not via router menu) and reset all settings (reset button on the back).