To answer your question, no you should not. RFC3927 in Section 1.6 prohibits this type of use.
Specifically, the last paragraph of this section says this:
Administrators wishing to configure their own local addresses (using
manual configuration, a DHCP server, or any other mechanism not
described in this document) should use one of the existing private
address prefixes [RFC1918], not the 169.254/16 prefix.
This rules out the whole /16 for this type of use so you need to look into a different alternative.
My first suggestion would be to use a loopback interface. Loopback interfaces are perfect for communication between services within the same host that do not require access outside that host. They are used in this way by a number of services, for management interfaces, testing and other purposes.
You mentioned in your comments/edits that the appliance will not let you do this. You don't mention the vendor/model or code versions, so my first recommendation is that you contact the vendor. If this is truly a valid use of the device, they may be willing to adjust their code to allow the use of a loopback interface; they simply may not have considered this use case when writing code to validate IP addresses. Or they may tell you why this is a bad idea and why it should be done another way.
If a loopback interface is truly out of the question, then you should use RFC1918 address space for this purpose. Make sure you work with any relevant IT staff is selecting the IP range to utilize to avoid any other unforeseen problems on the network.