Signatures in iMessage
I'm not sure it can be done automatically, but...

I'd just use the shortcut method. 

I use 'mynm' to trigger mine - it's not something I'm likely to ever use for another purpose. Same for myeml myphone myaddy etc

Substitutions can even do emojis or multiple-line text with Return key - so long as you copy-paste them into the shortcut, not try to type them 'live'

You'll find it's easier, long-term, to type a few letters at the end of your message than fight it every time you don't want it to happen