Currently, it is difficult to get the device to open up the captive portal, meaning I would need to use it on my phone. However, even after completing the captive portal, some systems rely on the IP address for authentication tied to the MAC. However, it doesn’t work on GL.iNet routers because it does not have a DHCPREQUEST input (Ability to have an option to input an address, such as 172.16.95.200, in which the router requests for this address without the issue of static IP addresses) or sends a DHCPREQUEST packet, which means the upstream router assigns a new IP address, requiring me to pay for the service again or use a static IP address if the upstream router accepts it.
If I run with a static IP address, eventually another device may claim the IP address if the lease runs out, which results me into still having to pay again for the service. Manually setting it in LuCI resets for each time the router connects to the router.
That is only the case if your MAC changed or the lease is already expired.
DHCPREQUEST input does not make any sense, imho. It's completely against the way of how DHCP works. 
Yes, I agree, but some public networks at least in my area, a lot of these don’t link the MAC to the IP address assigned, and their captive system relies on the IP instead of the MAC for some reason.
I am confused because DHCP requires you to link MAC to IP.
So the easiest way to solve this issue would be by setting a fixed MAC, no?
Yes, but some older routers assign another IP to the same MAC after the connection drops unless requested back (which is what most devices do), requiring me to use a static IP but with the risk that the router doesn’t accept static IPs or the lease expires