Can the word 'have' be omitted without changing the meaning here? [closed]

You say this:

So I argued that it had to be "and have become the recommended dishes."

But that's what the sentence already does say, you are just not parsing it correctly.

It's a combined version of two independent clauses:

• The recipes have been passed down.
  The recipes have become the recommended dishes.
• The recipes have been passed down, and the recipes have become the recommended dishes.
• The recipes ^{have been passed down} and ^{have become the recommended dishes.}
• The recipes have ^{been passed down} and ^{become the recommended dishes.}

In short, the phrase the recipes have, including the auxiliary verb, applies to both sections of the sentence. It's just that it's a shortened method of expressing it.

The following would be the incorrect version you're thinking of:

✘ The recipes have been passed down and the recipes become the recommended dishes.

Now, whether it's stylistically or idiomatically acceptable to omit the second have is a matter of opinion. At least as far as you're concerned, it seems that omitting it does lend itself to this kind of parsing difficulty. So, you're at least right in saying that it can be awkward.

This is similar to the following:

I struggled to pass the test and drive home.

It means:

I struggled to pass the test, and I struggled to drive home.
I struggled to ^{pass the test} and ^{drive home.}

It does not mean:

✘ I struggled to pass the test and I drive home.