When "etc." is followed by "that", does it still require a comma?

What you do is a matter of style. There are a few choices:

1. Use a comma, even though you normally wouldn't. This might be to avoid the appearance of the sentence ending—but then continuing.
2. Don't use a comma, because you normally wouldn't, and ignore the fact that it looks a bit strange in the context of the sentence as a whole.
3. Rephrase the sentence, replacing etc. with something like and so on.
4. Use the actual non-shortened word: etcetera (or et cetera; unfortunately, this is another style choice).

To give a specific example of just one style guide, here is what The Chicago Manual of Style (17th ed.), 6.20, says:

The abbreviation etc. (et cetera, literally “and others of the same kind”) and such equivalents as and so forth and and the like are preceded by a comma; they are followed by a comma only if required by the surrounding text. This small departure from the recommendation in previous editions treats such terms as equivalent to the final element in a series (see 6.19). (According to a more traditional usage, such terms were often set off by two commas.) In formal prose, Chicago prefers to limit the abbreviation etc. to parentheses, notes, and tabular matter …

      The map was far from complete (lacking many of the streets, alleys, etc. seen in earlier iterations).
      The philosopher’s population studies, classic textbooks, stray notes, and so forth were found in the attic.

            but

      For a discussion of periods, commas, and the like, see chapter 6.

The abbreviation et al. (et alia [neut.], et alii [masc.], or et aliae [fem.], literally “and others”), whether used in regular text or (more often) in bibliographical references, should be treated like etc. If et al. follows a single item, however (e.g., “Jones et al.”), it requires no preceding comma. (Nor is a preceding comma required in the rare case that etc. follows a single item.) Note that neither etc. nor et al. is italicized in normal prose (see the first example above).

If you strictly follow that guidance, etc. wouldn't be used in the sentence in the question at all, because it's not present in parentheses, a note, or tabular matter. But if you were to use it, the guidance would be to not use a comma, since the surrounding text wouldn't normally have one.

However, you might not choose to follow the Chicago guidance.

There is no universal rule. Pick one of the above approaches and apply it consistently.