What should I focus on when asking for conference travel support?
Its better to be honest and upfront. If not for ethical reasons, then for logical reasons that networks collide and you don't want to be found guilty of a few hundreds/thousands of dollars - which in the long run would be not much.

I was in a similar situation. I sent something similar to what you wrote and the folks replied back and gave me the funding from my new location. The other way to do this would be showing your affiliation as that of the "/". This is a technical fix - since that should get you the benefits of both locations. I prefer this because I am honest about where the work was done, and where I am today. 

Hope this helps!