I sounds like the system is set up against you. Doesn't AppDev have the same problem? What is different?
I'd have a conversation with who you report to - or the next level up - about this problem. If they shrug their shoulders, they are telling you that either (a) testing is chronically under-funded, (b) Testing is funded at par and the company never invests in improvement, or (c) Test leadership lies about the needs to get more $$ than it needs for the projects to invest in improvement.
(c) does not need to be a /lie/; it might be that test determines the test effort on a project then adds 0.20% for overhead, and that overhead pays for improvement.
I guess I don't understand enough about the staffing model to make a strong recommend. If I understand correctly, it if were me, I'd likely try to establish a baseline HQ team and a few team leads that were employees, then use the discretionary $ to fund contractors. I mean, your salary doesn't need to be justified to a project, right? Expand that sphere of people we "just" pay for every year.
Finally, again, I think the system is stacked against you. The ROI and justification route is going to get you answers like "not this year", or "yes, but ..." the cards are already dealt. I would spend my energy breaking out of the system, not trying to work within it.
Will you be at STAREast? Perhaps we can chat then.