Colin
(Sorry about the DHL man interrupting your call the other day!)
I get the impression that the boat has been similarly equipped (i.e. with the same ESC, DB etc) before this problem started, and that the only recent change has been the fitting of new batteries.
I reckon you've either not charged up the batteries properly or that they are somehow faulty (unlikely from new). The symptoms you have described would fit perfectly with that scenario i.e. the batteries quickly fall below a useable level under load; will not respond immediately when switched off and on, but recover some charge after "resting" for a while. The radio will stop working only if the regulator on the P92 is fed with less than 5.5v, so the batteries must be really flat!
To check the integrity of the installation remove the two new batteries and fit a known good 12v battery (fully charged), then run the boat on the water. If it goes happily for a while then the problem isn't anywhere in the electronics.
Are you using a charger sufficient for 12AH batteries? Did you charge them for long enough before you fitted them? I'd recommend a full 24 hour charge each, using a proper constant-voltage SLA charger of at least 1000mA output. If you have connected the two batteries in parallel without some form of diode then one may be back-charging into the other, with the result that both will go flat in fairly short order.
Keep us posted.
FLJ