I'm so sorry that Willow's having a tough time. It's so scary when they're ill and you don't have a straightforward diagnosis
I think you're doing the right thing having her referred sooner rather than later. She definitely sounds like her neck is painful too:( I think having the thoracic radiographs done is a good idea at this point if the antibiotics haven't cleared up her cough after this length of time and the bloods are showing an active infection. There are other tests they can run to try and culture the actual causative organism too so that the right antibiotic can be used. Poor girl she's having a really rough time
Hugs to you both and fingers crossed she's on the mend really soon.