The alien did a lot of things to technology that our heroes either found odd or then did not, without rhyme or reason, and supposedly because the alien was clouding their thinking. Among the former was the changing of modern weapons to swords and making the bulkheads impenetrable to cutting tools - the heroes considered this very mysterious. Among the latter was the original dropping of the isolation bulkheads and the locking of the ship to her amok course - the heroes simply assumed the Klingons did that somehow, even though such an assumption was irrational and probably wouldn't have occurred to them without the influence of the alien.
But as for intra-ship beaming, this is the dialogue on the issue, long after the alien influence had turned everybody into idiots:
Kirk: "We can't get through the Klingon defences in time, unless... Spock! Intra-ship beaming from one section to another. It's possible?"
Spock: "It has rarely been done because of the danger involved. Pinpoint accuracy is required. If the transportee should materialise inside a solid object, a deck or wall-"
Scotty: "Even if it could work, she may be leading you into a trap!"
Kirk: "We're all in a trap. This is the only way out."
Spock says it's possible and has been done; Kirk knows about it, even if only vaguely. It's clearly a desperation move, and in normal circumstances Kirk would probably much rather cut through walls. But the walls don't allow for cutting this time, and Kirk isn't normal himself: the alien presses its victims to take deadly risks, as it loves the resulting excitement and carnage.
Basically, we could rather easily argue that intra-ship beaming never was a very viable option in "Naked Time", and a dozen other things would have been tried before it, and were.