OK, I missed that the alarm goes away after some time.
And you said the alternator is two wire only, so + and - going to the battery, and nothing else, right. That means it's a completely self-contained regulator with automatic on/off switching.
I'll bet this is all really very simple. The new alternator just has a longer delay before it turns on, yet the engine alarms are expecting to see a voltage increase right away, or very quickly after starting. In time, the new alternator turns on, starts charging, voltages rise, and the alarm goes away.
David's suggestion of putting a meter on and watching the voltage from start up until the alarm goes away will confirm or deny the theory.
If the theory pans out, then you can either ignore the alarm, or see about changing the alternator or regulator to something that comes on right away, or is controlled by the ignition switch.