How much is he drinking at lunch time? Cause I might limit his lunch time consumption of fluids cause that might be the difference in that he is being allowed to drink way more right before nap at your program than at home which is why he can make it through nap time there?

As for putting a pull up back on for me it would depend on how the 'accidents' are affecting the program ~ is he sleeping in a spot that the pee is not posing a hygiene risk or damage to flooring/bedding/etc .... are they easy and quick to be able to clean up for you?

Cause my first concern would be ruining the furniture where he sleeps so in the interest of hygiene and dealing with clean up of bedding if it was something occurring 'regularly' I would putting a pull up back on OVER TOP of the undies as a containment unit ~ so if he has an accident he will still get wet undies and feel the accident but it will not cross over onto bedding or floor? If it is just something 'rarely' occurring 1 or 2 a month than I would put a pee mat on his cot area to absorb and accidents and not use a pull up and really reflect on what was 'different' that day in the way of consumption of fluids or STRESS cause either could be causing the sleep accidents.