Great! Do you mind sharing your schematic? :)
You are right about the ROM editor. It should support pasting, or some other form of input (e.g. load external files). In case of pasting the problem is the format of the string/data. The problem with loading external files is that a native open file dialog won't work correctly in fullscreen mode, and writing a full file browser with the current GUI is going to be a PITA :)
Until I figure it out, you can work around the issue by using a scripted component for your ROM as I did in the 6502 schematic for the Kernal ROM (https://github.com/jdryg/dls-schematics/tree/maste...). Either extract the component from the schematic or copy/paste the script into a new component. Then, Base64 encode your ROM and paste it into the script. It should work. You can even make your compiler generate the whole script and just paste the output into DLS.
Hope that helps for now. Thanks for the feedback!