whatever that

I wouldn't often insert "that" here.   But one of the many functions of "that" in English is to wrap up the verb phrase that follows so as to package it as a noun, and it's so general that I'm not sure it's ever actually wrong.   Some examples:

Have you heard that they're coming tomorrow?

It must be that she's confused.

Will you please see to it that they buckle their seat belts?

The idea that we're in love is totally ridiculous.

It's hard to admit that I was ever wrong about him.

That I was ever wrong about him is hard to admit.

I believe that you won't find her until it's too late.

In most of the above examples we could omit "that", but it would still be understood.   And in some older usages "that" is used where we now never insert it, after some adverbial prepositions for example:

I believe that you won't find her until that it's too late.

I must leave before that I become too attached to this place.

So I wouldn't say that your second example is actually wrong, not as who should say wrong.   But I would never say it that way.   And I wouldn't say[that] it has a different meaning; I think [that] they both mean the same.

I say that the "that" in the original question is flatly wrong. It is a foreignism. I'm not an expert on why, but I will try to find out.
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He eats whatever is sweet.

He eats whatever that is sweet.

In the example above, whatever is classified as a compound relative pronoun. The compound relative pronouns function simultaneously as the relative pronoun and its antecedent. Other examples of compound relative pronouns are what, whoever, and the archaic whosoever, whatsoever, whoso. These archaic ones might remain familiar through their use in the King James Version of the Bible.

In all of the examples above the that is a relative pronoun.

When using these compound relative pronouns, you cannot add another relative pronoun such as that. There is no need for it. It's wrong.

Say what you think. (and not: *Say what that you think.)
I like whatever you write. (and not: *I like whatever that you write.)
She fires whoever disagrees with her. (and not: *She fires whoever that disagrees with her.)

In these uses as relative pronouns you should be able to replace them as follows:
what = "the thing that"
whatever = "anything that"
whoever = "anybody who"

Not to muddy the water, but you have to be careful to distinguish between that as a relative pronoun and that as a demonstrative pronoun or adjective as in the following correct sentences:
I asked her to document whatever that does.
He obeys whatever that woman says.

In the above sentences, the that is functioning as a demonstrative pronoun or adjective, and not a relative pronoun.

Years and years I have wondered about this issue. Finally -in black and white

Thanks a whole bunch, Glenn.

Bravo, Glenn.   I never learned about compound relative pronouns in high school English, but you explained it perfectly.   I am now wondering about Bob's list.   Going strictly by how it sounds to my ear - everything in the list sounds okay to me except the last two (adverbial prepositions?) Can anyone tell me how "that" is used in these examples. Bob talks about what it does but he doesn't define it's use.