My instinct about it is that it is originating through an interaction with your other audio settings somehow. On a modern device (any device that can run iOS 8.1) I would expect that to take about a second and probably including the period of the pause.
Normally at this point I would ask you to try to replicate it using the replication technique so I can take a look:
https://www.politepix.com/forums/topic/how-to-create-a-minimal-case-for-replication/
That is basically taking the sample app, replacing its vocabulary with your vocabulary, and adding the RapidEars methods (but not adding any of the other features of your app). My guess is that you won’t see the same lag in this case, but if you do, you can then make a recording of your session using SaveThatWave so I can replicate the same session and see what is happening. There is more info in the forum post I linked to. I think this would be a good next step because it looks like there are many things going on in your app session which have the capability of interacting with the audio requirements for OpenEars, so it’s hard to give troubleshooting advice on the app with all the other features integrated.