If it's after she's been chasing something and running flat out, even if only for a short time, it sounds normal especially if she stops when she has a rest. I think Jackson Galaxy actually says cats should be panting after a good workout? If there is a change in how quickly she starts to pant then that might warrant a trip to the vet. Has she panted before or is it just this particular toy that gets her running out flat so that she pants? They can pant quite quickly if they run up and down chasing something and sometimes they find a toy they really love and just go flat out running after it.
Definitely keep the string away from Barnaby. I left da bird rod lying around for just a few moments during a play session with Mosi when he was little and he chewed through it. He vomited up some of it (including the clasp) but quite a bit of it was unaccounted for so he had to have surgery to look inside in case any was tangled up in there.