Connecting to an Unknown DHCP provided IPv4

Projects Networking, Protocols, and Devices
1

This question is out of my depth by a huge margin so my understanding is very limited but effectively Im using an Arduino with an Ethernet shield to host a web server that is essential for setting up the device (some config values specifically). The only issue is the device itself has no way of displaying the IP that the device connects to by default, meaning I need to either use a static known address for the setup page, or find an alternative method of communicating the DHCP provided IPv4.
Is there any way for me to tell the router "redirect any request to blablabla.com to this ipv4"?

or is my only option to just hope and pray that the chosen ipv4 is open so that we may tell the end user where to connect to?

2

You can setup your router to always give the same IP to your device using the MAC address of the Ethernet shield. Basically you map the MAC address to an IPv4 address. Now if the router doesn't support that or you have no control of it, then you are out of luck