Are Benadryl (diphenhydramine) allergy dissolving strips usually safe to use while having bronchitis?

Yes. Yes they are generally safe, but may not be effective or recommended. Older antihistamines such as Benadryl (diphenhydramine) could be too drying when you are experiencing bronchitis. If you are having bad allergies on top of the bronchitis, try a newer otc antihistamine such as Loratadine or cetirizine.
Careful... You need to be careful that you don't dry out the cough too much making it more difficult to expectorate the sputum, especially when it is acute bronchitis with infection present. You don't want to keep infected sputum in your airways so use very little Benadryl (diphenhydramine) or cough suppressants in this case.