Is there a means to contact Apple support using message?

Have you tried downloading and using the Apple Support app? It's often easier than going through the web interface. If I recall, you end up chatting with them through Messages. But it's been a long time.

See also:

Genius Bar Reservation and Apple Support Options - Apple

Not that I am aware of. It's either chat or a phone call.

I was on chat yesterday and was told that someone would call me in 5 minutes. No call ever came. I finally remembered the passcode after 24 hours of trying everything that I thought I would use as passcode for FaceID. Ultimately I didn't need it after all.