@panteon00
Your answer seems to be quite simple, your WiFi router is getting the IP 10.0.0.2 just because you have connected the Ethernet cable from Router to WiFi router directly. I can suggest two things which you can try in your case.
Solution 1:
You can just plug off the Ethernet cable from WiFi router and then plug it on your client machine. You will automatically get your desired IP.
Note: If you do so, your other devices won't get Internet connection as you have plugged off the Ethernet cable from WiFi router.
Solution 2:
You can connect a switch between your WiFi router and the service provider's router. Then connect both your WiFi router and the client PC on switch. And you will get the desired IP.
Note: You must take permission from the owner of router before implementing solution 2 as you will be consuming multiple IPs from their IP stack. Most of the times, if the Router is owned by your ISP then they won't allow you to use a switch.