Wow. Your response is amazing and very impressive. Even if you are not right, it's very interesting to know. Thank you so much!

I think the grated, matrix-lattice-like sign may have something to do with water, that it signifies, indicates, or means the availability of drinking water at the Tavern, Inn, Hostelry, or Pub. When such Taverns first emerged in medieval times, there was a law passed that in exchange for them having the right to serve alcohol, they were obliged to provide a free tankard of water to anyone, any passer-by or traveller, who requested a drink of water. And, as we witness in the film, a tankard of water is what Redmond Barry requests from the woman at the Inn.
The origin of such a symbolic motif, of such a design indicating water may have some connection with the shipping and maritime world, as all ships had such grated designs on their decks serving both as doorways and as the decking itself, the purpose of the grated holes in the doors and decks being to allow water flooding onto the deck to pass through the holes and off the deck again and avoid any build-up of trapped sea-water.***
So the aesthetic design of such a symbolic sign is long connected or associated with water.
They were all later replaced with more elaborate and seductive Tavern signs, with paintings, calligraphy, and ornate carpentry to entice the traveller to enter the hostelry. One of the earliest forms of 'billboard advertising'
***Here are some images of vintage wooden ship gratings, all of which look remarkably similar to the Tavern's grating signage in this scene in the film:
http://www.skipjackmarinegallery.com/product/DCG232.html
And this site provides more examples, in addition to describing how they are constructed:
https://dovetails.wordpress.com/2009/10/21/the-versatile-grate/

Not lettuce, lattice

You chop it up and eat it with "dressing", sometimes with other vegetables such as tomatoes, onions, cucumbers, etc.
It's still very popular, I'm shocked you've never known of this.