They likely could have, but I imagine they left it alone on purpose.
For starters, that is one of the most beloved episodes of the original series; messing with the original footage at all would likely not have gone over well with fans.
But beyond that, I suspect they were using it as a chance to clear the issue up. Ever since the first time we see Worf on The Next Generation, his features had been a matter of debate among Star Trek fans. The fact that original-series-era Klingons did not look anything like Klingons from subsequent shows was left unexplained, at least in-universe.
The idea of some sort of genetic engineering or similar explanations had been floating around unofficially for many years. In that particular episode of DS9, the writers decided to finally address the question on-screen. Though they didn't really explain anything, we at least got confirmation that the change in appearance was canonical. Unfortunately, we would have to wait until Enterprise to get the full story