Severe ARP flooding from PC requesting link-local address

Partial answer:

• One request every second isn't "flooding".
• DHCP doesn't assign link-local addresses, nor is it involved with link-local addresses in any way, so you can't fix this via DHCP.
• ARP requests are not sent "to the router", they are broadcast to everyone in the LAN segment. The computer with this address is supposed to respond.
• If your PC sends ARP requests, it's either looking for someone with 169.254.255.255, or it wants this address for itself. So the first step is to check which addresses are assigned to this PC.
• If the PC wants 169.254.255.255 for itself, disable assignment of link-local addresses to interfaces (details depend on OS).
• If the PC wants to communicate with 169.254.255.255, find out which application wants to do that, and stop this application from doing so.

I have limited leases for the precise number of devices and i know what each IP is

Again: Link local addresses are not managed by DHCP. They don't have leases. Each device is free to assign itself one address.

I check my router logs and theres no Mac address or device associated with the link local address.

Of course, because the router doesn't keep track of link local addresses.

Not sure what application has the authority to send ARP requests

Anything using the network stack can send out ARP requests (which is handled by the OS). On top, the OS can assign itself link local addresses (to repeat, without having leases, and without asking the router), and that will cause a broadcast to verify if this address is taken.

Does this mean there IS a device currently using that link local address

It's possible. It's a link local address, there's no track of which device assigned itself which address. The only way to find out is to log in to each and every device, and check the assigned IP addresses.