I agree with Donna. The day trip is totally do-able. I had the same issue trying to book a direct train well in advance. By the time those tickets were on sale, most of the recommended tour agents -- Overlord, Dale Booth, etc. -- were booked. We went with Normandy Landing Tours.
https://normandy-landing-tour.com/
It's not the slickest operation, but we were perfectly happy with the welcome we received from Jean Marc. Guide Ramon took about half a dozen of us around to a few of the major sights -- Arromanches, Omaha Beach, the American Cemetery, Pointe du Hoc -- efficiently and knowledgeably. It was definitely a quick-hits-highlights-tour, but that's what we had time for.
Upon arriving on the morning train from Saint Lazare to Bayeux, we made our way to the nearby Hotel de la Gare, where Jean Marc assured us we had time to walk into town to see the Bayeux Tapestry before the tour. We were so glad we did, as the Tapestry is one of the most well-presented artifacts I've seen, and the town is lovely. We walked back to the hotel, had a quick lunch and were off on the afternoon tour. Afterward, Ramon dropped us off near the Bayeux Cathedral (Our choice. We could have gone back to the hotel). We had time to grab a gelato, walked back to the train station in plenty of time and were back in Paris for a final dinner before flying home the next day.
As many have said, it would have been wonderful to have more time to see these sights, but we had only one day and I'm very, very glad we spent it this way. It was for certain a long day, but most worthwhile.