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The short answer, i would say it is never a good idea. Mainly, because even if the dog isnt aggressive or fearful, any punishment done with the hands can lead to fear or aggression associated with the same movements.
In this case it may cause problems when people reach under the dogs to either pet under the chin or reach for the collar.
Unless, the mouthing is causing injury and not just ‘annoying’ such as jumping, pulling on the leash, and not obeying in general it is a behavior ideally addressed by a training plan which shouldnt be that difficult and can include different types of punishment in the process, but it seems as if the shelter may have your hands tied.
As far as i know none of the major dog training professional organizations stand by the fact that tug makes dogs aggressive anymore than chasing a ball will make the dog want to chase a child.
I would suggest talking to whoever makes the decisions there, because if they dont mind a hit under the muzzle but do forbid tug you and the dogs will have an uphill battle.
Some things you may do in the meantime to manage the problem and break habits is to:
Bring a toy in the pen with you. Simply hand it to the mouthy dog. Sometimes they just need something in their mouth and you may be able to sneak a little tug before you release it to prompt them to hold it. Then bring to the exercise area and play ‘fetch’ with the toy.
You can also walk into the pen with high value treats or peanut butter on the end of a long spoon that you can hold out from the side of your hand while you buckle the dog up.
Once you get the leash on the dog it is easy to prevent mouthing just by using the leash.
If you need to punish earlier rather then later almost anything is better than the hands to prevent side effects. A leash correction, a blast from an air interupter, or even a high power citronella spray such as ‘spray shield’.
Some dogs you can get away with simply grabbing them by the collar with one hand and firmly stroking them with the other until they calm down before putting on the rest of the equipment.
Keep in mind that all training takes patience and for most problems it is reasonable to have a management phase as you work on the training plan to teach the better replacement behavior.
