This needs to be done by the property/customer's IT team as the main website is owned by customers.
The Web Messenger Chat's icon, as shown below, can be updated by updating the businessiconUrl field in the smooch script of the customer's website.
Within the smooch script there is a businessIconUrl field where customers can set the hosted image.
This is an example of what it could be set as:
<script>
Smooch.init({
integrationId: "66xxxxxxxxxxxxx565c",
businessIconUrl: "https://example/logo/icon.jpg",
businessName: "Reservations",
Either this URL would need to be updated to point to another image or customer should be able to update the image where its hosted with the same name so it keeps the same URL.