When you share the APK file to WhatsApp and you see BIN, it's actually a WhatsApp bug and has nothing to do with Flutter. Netizens have also reported that PDF's that was shared on WhatsApp are shown as BIN (Binary file), hence they are unable to access it.
Solution
Locate the APK file on your Android file manager Android/media/com.whatsapp/WhatsApp/Media/WhatsApp Documents/.
Share to file through other means, USB cord, Bluetooth, Google drive etc.
Use adb CLI to install the application directly on your phone; adb install path-to-apk