I'mCustomerand I would like to try and help with your question. I would not pursue a dental cleaning for Lilly under any circumstances until you have resolved your her issues with trouble breathing. You mention that you've been told she needs a $1200 surgery to help her breathe but you don't mention what that surgery is exactly. Also your vet recogizes the odor in her mouth but does not believe it is a result of dental disease. Where does your vet think the odor may be coming from? How old is Lilly? Age should be considered when considering the likelihood of the various types of problems that can cause difficulty breathing.
Without knowing more about your Lilly's exact circumstances such as what surgery your vet recommends to help with your dog's breathing and his theories on the source of the foul odor in her mouth, it is difficult for me to give you specific advice. If you could post additional information about what surgery is recommended or any diagnostics and tretaments Lilly has already had for her trouble breathing and/or odor from her mouth that would be helpful.
Until I have more information, I will tell you what I would suggest for a dog with Lilly's problems: she needs a full physical exam with special attention to her tracheal/pharyngeal region as well as her heart and lung sounds. In addition, the lymph nodes at her neck (i.e. submandibular region) and elsewhere should be palpated carefully. She needs blood work: specifically a full chemistry profile, complete blood count, urinalysis and a heartworm test. In addition, she needs 3 view chest x-rays that include views of her neck area. Please note that the incidence of heartworm disease is increasing in Australia. In addition, please note that the Cavalier breed is very prone to heart disease which could cause the trouble breathing that you have observed so your vet needs to be very sure this is not Lilly's problem.
In the event that Lilly's exam and all those diagnostics reveal no specific cause for her trouble breathing and the odor from her mouth, then I would recommend referral to a specialist where advanced imaging such as fluoroscopy, MRI/CT scans and/or bronchoscopy are available. If the cost of those specialized tests was unaffordable, then I would recommend a diet to maintain a healthy weight (i.e. obesity if present contributes significantly to difiiculty breathing of many causes), a trial on a cough suppressant such as hydrocodone and I would consider a trial on a bronchodilator as well if there is no evidence of heart disease at all. It would also be important to avoid exertion and overheating so Lilly would need to be inside in an air conditioned environment without stress for as much of the day as possible. Occasionally, antihistamines are helpful if part of Lilly's problem breathing is allergic in nature. Further presumptive therapy would include a trial on antibiotics such as clindamycin for the foul odor that has been noticed.
Please provide more information if possible about any diagnostics or treatments already preformed by your vet and let me know what this surgical procedure was that he recommended. If he hasn't recommended routine lab work, heartworm testing or x-rays, then get a second opinion. It is inappropriate to make surgical recommendations for difficulty breathing without doing the aforementioned diagnostics first.
I'll look forward to your response and hope that the information provided thus far has been helpful to you.