I'm now able to run bitcoind -daemon independently of Armory as long I immediately start Armory after I started bitcoind -daemon.
I guess that once bitcoind has established to many connections it doesn't accept RCP calls form Armory anymore.
Just peering through the fog
Don't really know what's going on there. The RPC connection should be available regardless of Core's p2p connection count. You could try to automate bitcoind from Armory to make sure the connection is grabbed early. In doubt, I would try to monitor the Core RPC port, see what's trying to connect through that.