IMHO Theoden basically didn't want to take Merry, a foreign guest who looked much like a child, into great danger.
Merry certainly could have ridden a full sized horse if he was three or four feet tall and knew how to ride.
Ex cavalry officer Charles King, one of the main inventors of westerns, wrote a novel Trumpeter Fred in which the title character was small enough to be picked up and put in the saddle,and yet rode the biggest and strongest horse in his troop.
The Photographic History of the Civil War in 1911 mentions little scamps of bugle boys who had to be boosted up into the saddle and rode in wild cavalry charges. A specific example is Jimmie Duggan of the 5th US cavalry who was fourteen years old and only three and a half feet tall in 1861.
Bugler Joseph Fought, who enlisted in the 5th Cavalry in 1860, was fourteen years old and four feet eight inches tall, much taller than a normal Hobbit but not that much taller than Merry was after drinking an Ent-drink.
A plains Indian boy the size of a Hobbit could have ridden any tamed horse, no matter how large.
Elizabeth Taylor, aged 12, filmed a scene where she fell off a horse in National Velvet (1944) and injured her back.
Arabian camel races until very recently had small, Hobbit-sized, boys riding racing camels at least as large as most horses.
Many Asian mahouts have ridden and controlled elephants much larger than horses when the mahouts were boys no bigger than hobbits.
Here is a link to a photo from Elephant Boy (1937):
Furthermore, if the Rohan riders were like medieval knights they wouldn't have ridden their battle chargers for hundreds of miles to get to the battle. They would have ridden smaller riding horses for days to reach the battlefield and then mounted their huge chargers to charge into battle.
Tolkien ignores the fact that many mounted war groups brought along several horses for each man, with the riders changing horses from time to time to keep from wearing the horses out. Many mounted war groups would also have other horses (or mules) to carry supplies.
So in many real life war groups taking one or two more horses to carry a child-sized person to the battle would be no trouble.
So either Tolkien was ignorant of such matters, being an infantry officer in World War I, or else he made Theoden ignore such options because Theoden doesn't want to risk the life of his small new friend in an almost hopeless mission to "ruin and the world's ending".