For other people with the same problem: I found a work around which on the first glance works!
Steps:
- Log into the WiFi network using any device
- Push the button on the captive page
- Write down the device’s MAC address for this connection
- Deconnect from the WiFi
- In the setup() of your sketch: Change your ESP’s MAC address to the address of the device before the blynk function starts, see: Get ESP32/ESP8266 MAC Address and Change It (Arduino IDE) | Random Nerd Tutorials
- Enjoy Blynk with a network using captive pages!
It is possible to assign a new MAC address to the ESP board: