E.B.
The first thing that came to my mind is that many antihistamines are drying - they can make your throat or mouth feel dry. Your daughter might be dealing with a dry throat, esophagus or mouth, and the cough might be a natural response to the dryness.
Secondly, take advantage of your pharmacist. Hopefully since your daughter has been on meds for so long you have a good relationship with a pharmacy. Ask the pharmacist to review your daughter's medications, including the changes and the new addition. Ask if any of those have dryness or throat irritation or a cough as a side effect. It could be as simple as giving your daughter a soothing throat spray (a non-medicated one, intended to lubricate or soothe a dry mouth), or perhaps a dry mouth rinse, etc. But if it's a side effect, the pharmacist can be your best source of reliable information.
Also, some medications can cause a slight heartburn that makes you want to clear your throat. It might not even be discernible as far as pain goes, but it might be enough to prompt an involuntary response, such as a little throat clearing or cough. Again, a pharmacist can help you there.
Good luck.