"Keelson" is the term for an inside keel. It will minimally add some strength and stiffness, at the cost of providing something that will perpetually be in the way -- something to trip over and to keep things (coolers, bait buckets, etc.) from laying flat on the floor. I wouldn't bother -- the folks who built the canoe didn't, and your canoe looks like they know what they were doing.
If you look around, you will see that keelsons are very rare in canoes -- when they do exist, as often as not they are present as part of a home-built or kit canoe, where the keelson served primarily as part of the construction apparatus rather than as a necessary structural part of the canoe.
A wooden canoe will have a certain amount of flexibility -- up to a point, it will bend rather than break, which is why they can last so long (one of mine, the one in my avatar, was built in 1922, and is still going strong -- the one below is from 1931 and had several broken ribs at the time of the picture). A certain amount of care should be taken -- they are not fiberglass or aluminum and won't take the kind of brutal abuse that such canoes can bear -- I have a Royalex canoe for when I am planning to paddle through rock gardens -- but they get their quietness and special feel and character in the water from their relatively light weight and bit of flexibility from the nature of the wood -- strong, flexible, sound-absorbing. Wooden canoes don't need to be babied --
but you do need to give them some respect.
Further. your photo above shows the canoe supported on two horses near the ends of its length, leaving the middle inadequately supported. In such a situation, you are going to see the canoe flex, and indeed, if you were to step into the canoe supported that way, you well might break something. But you have a canoe, not an I-beam, and neither it nor any boat should be expected to act like an I-beam. A canoe (and most any boat) is designed knowing that it will be supported by the water along its entire length. You would likely not see any flex at all when in the center of that canoe while it is floating, even with a heavy cooler or another person right next to you. It is probable that you are seeing your empty canoe flex as you lift it and move it around on dry land, especially on those horses. But with even a heavy load spread out normally -- people and impedimenta spread along the length of the canoe -- you would not likely see any flex.
Move those horses closer together -- generally try to have about 1/3 of the length of the canoe on either side of the canoe -- the center of the canoe, being wider than the ends, usually has more weight and support should be placed to take that into account. The stern of your canoe has a transom, so your supports might be
placed asymmetrically, with aft support might be a bit closer to the stern, to deal with the slightly heavier stern.
The wood/canvas canoes shown below were loaded for a week-long trip on the Allagash, with each canoe carrying the personal gear of the paddlers and sharing the common items -- cast-iron cooking gear, coolers, food, etc. -- for 12 people and a large dog. The water was shallow, so the canoes were dragged across gravel bars at times and occasionally encountered rocks, and gravel landings at camp sites were usual. Three of the canoes on this trip were wood and three Royalex -- the canvas-covered wooden canoes (not fiberglass covered) fared quite well, as well as the Royalex -- and not just for a rare camping excursion -- these are boats used hard by professional guides all season long, season after season.
I think you are underestimating the strength of wooden canoes. Don't try to make it something it is not (a plastic or aluminum canoe) -- respect it and use it and enjoy it for what it is, a type of boat that has given service and pleasure for well over a century.