Any email provider can track you pretty successfully whether web based or using another protocol such as IMAP. Most email is at best protected by encryption only while in transit after all. For personal email you get to choose your email provider and whether you are ok with them tracking you or trust them not to track you.
But an example of a non web based email client which provides privacy protections regarding images in email is Apple Mail and its mail privacy protection features.
Whether Apple Mail supports BIMI is not really relevant since my original comment was regarding email.platforms supporting caching of images and not BIMI specific. If an email client supoorts caching of images extending that to also include BIMI logos while adding BIMI support is minimal effort.
That being said Apple Mail has supported showing BIMI logos since iOS 16 and macOS Ventura. Do they use caching for doing so when mail privacy protection is enabled as they do for other images? I have not specifically done an in depth dive to determine but what exactly would the motivation be for Apple to bypass the image caching functionality for just this type of images?