Creating a project from the SDK for the LPCXpresso54628 board allows me to create a configuration using the blue debug button, and I can go into a debug session with no issues.
If I modify that generated configuration to work with my makefile based project and copy it to the project, after launching it I get the following error:

I also tried making an empty project from the SDK targetting my desired processor and could download it and debug it into my custom board. I decided to try and edit that configuration to use it on my project but that one also got the error above once I got to actually trying it out.
If I hit the blue debug button to generate a configuration on my makefile project (the one I'm actually trying to debug), after I get prompted to select a connected probe, nothing happens. I know this is not working right because if I do the exact same process with a project created from the SDK I can actually generate a debug configuration using that button.
If I got through the green button and try to generate a new configuration from scratch, I get the following:

This happens just after double clicking the GDB SEGGER Interface Debugging option on the left panel to create a new configuration. If you hit yes it lets you edit values but once you get to actually trying it out it goes into the loop I described in my initial post.
For reference, I can debug this project on a vanilla version of eclipse utilising Eclipse CDT tools (GDB SEGGER J-Link Debugging option), I am just having issues with the debug session hanging when the RTOS is running and I try to halt the session and restart it. This is why I came back to MCUXpresso to try out the debugging here, but the project is the same, so there is nothing inherently wrong about it, the code works and can be debugged.
Could there be something about my project, which is not an SDK project, that causes these issues?