Try this simpler approach.
Ask the other person to send you a short message first.
If their iMessage arrives and shows Delivered on your phone, your connection and their phone work.
Then reply to that message.
If your reply still does not show Delivered, switch to SMS by holding your message and choosing Send as Text Message.
If SMS goes through and iMessage keeps failing for that contact, remove the conversation, restart your iPhone, then start a fresh chat.