I’m more than satisfied with Thunderbird and it’s minimalist design, but for receiving notification its window has to be open, which makes my active windows, a tiny bit crowded. Could it be anyhow forced to run in the background?
Try systray-x from aur.
xfce4-mailwatch-plugin to show an icon when there’s new. the mouseover will show a count broken down by account and i picked a fully transparent icon for when it’s zero so there’s no icon visible. you can set a command to run on click, on first new message, and on any change of message count, so you could set it to send a desktop notification on count change.
birdtray, available in AUR might be an option