What's wrong with using a clause starting with 'how' as the object?

It seems that the use of "how" here is still considered grammatically substandard in some quarters, whereas "the way" is universally accepted. Therefore you might want to use the latter in more formal settings.

It is certainly true that "the way" is the older and more established usage, though "how" has become increasingly acceptable since 1980, when it was barely used.

"I don't like how he looks" raised no eyebrows among either American or British speakers on WordReference.

In my own BrE circle "how" is widely accepted, and a quick google of the New York Times suggests that it is also common currency in AmE: They don't like how he's doing his job [2010]. I just don't like how they fit on me [2015]. I don't like how I appear in the mirror now [2020].

Nevertheless, the Cambridge Grammar 2002 marks it as "grammatical in some dialects only". Whether this is the case in 2020 is, to my mind, dubious.