There are few gender neutral honorifics in English that are going to sound completely natural in a conversational context, especially a call center.
If unwilling to work around the matter as others have suggested this may be a case where you could experiment with setting precedent for new cultural norms.
I have, on occasion, used the phrase my friend for this situation. However, that may be taken as too familiar depending on the circumstances. For example, I am quite put off when a politician with whom I have no other relationship addresses me as "my friend." Or even if this is done collectively as "my friends" when I am part of his audience. Yuck.
Hotels in their writing sometimes use the expression honored guest. Businesses more generally might adopt valued customer, esteemed patron and so on.
There's no grammatical reason these written forms of address can't be impressed into verbal service. And that they sound strange is largely a matter of becoming accustomed to them.