Technically this is possible with scripting. You have an automation that triggers every night to overwrite all text that says “Today” with the current date.
Having it update in real time while the user is typing is possible, but results in lost data if the user is still typing. See other example here: "Find and Replace" for rich text in automations + scripts
What I could do is edit the Better Dates Integration so that the date that is today has a “Today:” prefix (As well as a “Yesterday” and “Tomorrow”. And maybe some other stuff. Then you can mention that date, and it will perseve the date even after it’s no longer today.
That said, I think this topic fits more here: Today's Date Shortcut than here.