Seems you could use a merger, like this one:

http://www.midisolutions.com/prodmrg.htm

to merge the outputs from the Livid and Ar into the A4. The AH could then get sync from the A4 through port. (If i get your drawing right).
The merger requires no psu nor batteries. I use it for hooking a midikeyboard to the A4 with the AR as master.

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