Imho-ALL humans like to know where they are in a conversation. This isn't something appreciated especially by Japanese. My point is that there is no value add for being so forthcoming to the tencho over the phone.
If your language skills aren't up to the minimum language required to ensure that you can safely play, he'll know it right away.
Imagine if you were a tencho, and someone who didn't really understand English called you to reserve a girl. You could know very easily whether or not the guy could safely play with your girls by asking just a couple of questions.
My point is that you stand only to lose by explicitly saying you are a gaijin over the phone. Specifically, the girl selection may drop off a cliff, or you will get cockblocked, when the girl may actually have no problem with you.
I suppose that if a girl goes to your door, and is shocked to see that you are a gaijin that she may turn heel and run away, but I have never had that happen to me. In fact, the opposite has happened numerous times.
If anyone has had a girl run away after seeing the 'truth' of your appearance, I would like to hear about it, and would quickly retract my advice if so.