Many thanks to everyone who has replied, and apologies for not thanking you all earlier.
What a minefield this all is!
I must confess I am very tempted now to go in on a visa waiver. There are in fact two of us, one with an ICAO Com, and one with an ICAO ATP. As I see it, 'training' for the ATP is category 4 as far as the TSA is concerned, and therefore 'recurrent training'.... and no threat assessment is performed, and no fee charged, although some reduced information is provided to the TSA before beginning recurrent training. This is done after arriving but before beginning any training. The examiner is going to need the TSA number. Thanks Unfazed for your experience on this.
As we hold foreign licences it seems we need to get these confirmed by the CAA of the country of issue. Thanks for pointing this out Skyworker. I have looked at the legislation and this does seem to be the case. This, of course, just adds to the delay. Has anyone gone over without the confirmation?? It is the first I had heard of it! What about you jab?
Thanks also to Just Boxes, slim_slag, jab, and the rest. All your advice and comments, and the time you have put in helping me, are very much appreciated.