"I would like that he was normal"; is this sentence correct?

'I would like it that he be normal'. (Use of infinitive instead of inflected 'was' form flows better).

The other constructions with greater modification sound nicer, but this suggestion provides improved coherency at the most minimal amount of modification (one word changed).

Personally, I would go for 'I would prefer it for him to be normal', in written context, but in spoken context I would most likely say 'I'd prefer him to be normal'.

• For reference, I am a British English speaker (native).

The first form sounds as if you are mixing present and past tenses incorrectly. It's actually unclear because for each possible meaning you might want to convey there are preferred forms which are different from the one you give.

I would like him to have been normal. (Present feeling about past state)
I would have liked him to be normal. (Conditional past feelings about ongoing/current state)
I would have liked him to have been normal. (Conditional past feelings about past state)

and so on.

The second form is more natural, perhaps because of the conditional form.

Usually these sort of feelings are expressed more directly in the present tense

I would like him to bathe before going to sleep. (Implies he doesn't)
I like him to bathe before going to sleep. (Simply expresses a preference)

Unless you want the reader to understand how your feelings have changed, it is best to use the present tense.

If everything was in the past, just use the past tense

I wanted him to bathe before going to sleep.