It seems risky but you could try to modify the addinRegistry.xml file, by default it lives in the folder below. I think you could reference that file path by stripping the /Addins/ folder off the end of ADDIN_HOME(com.your.addin.id).
Maybe a safer solution is to check if the required add-in is registered in the load section of the dependent add-in, and then either load it if it isn't already, or run files with whatever functionality you need.
Generally I try to make add-ins self-sufficient but that does mean I have some scripts that I include in many different add-ins. For example to update a logging feature used in many add-ins, I update the logging script file, then copy/paste it to many add-ins and re-build/re-distribute those (or let the add-ins check for updates automatically).