I own an Avermedia Extreme Cap U3.
I can make it work beautifully with Windows 7, 8 , 10.
Resolume can also recognize it as an external video input, but no matter what I do, whatever I did so far, Resolume only captures 1/4 of the screen, meaning the top right corner of the input source.
I don´t know exactly what is your problem and what you have tried already.
You may get it to work or not, but there´s currently no official support from the Resolume team.