I want to say "My favorite animal is..." and then give the animal. But "animal" is neuter, so I'll end up with a predicate nominative that doesn't agree in gender with the subject! "Meum dilectum animal est avis". Is this possible? What's the solution?
The gender of a predicate noun can differ from the gender of the subject
There is no problem with a predicate noun having a different gender from the subject. Predicate adjectives are grammatically required to agree with the subject in gender and number, but predicate nouns are not required to agree in either of these categories.
In some cases, a noun of a certain gender might be inappropriate as a predicate with a certain subject for reasons other than gender agreement. Some nouns have meanings that are specialized to male or female beings, and in that case, you should use the appropriate noun. For example dea is a noun (grammatically feminine) with the meaning "goddess", while deus is a noun (grammatically masculine) with the meaning "god". Just as it would be false or a mistake to say "Mars is the goddess of war" in English, it would be false or a mistake to say "Mars est dea belli" in Latin. But this is a mistake of word usage, not of grammatical gender agreement.
Other nouns such as animal can refer to male or female beings, but only have one grammatical gender (in this case, neuter). Nouns like this can be used freely as a subject or predicate with another noun of a different grammatical gender. For example, "Homo est animal bipes rationale" is a grammatically correct Latin sentence that gives a well-known proposed definition of the word homo, a grammatically masculine noun meaning "human" or "man" (not "man" as opposed to "woman", but "man" as opposed to "god" or "beast", so not exclusively male in meaning).
Your sentence: "Meum dilectum animal est avis"
Based on the above, I feel confident that "Meum dilectum animal est avis" has no errors of agreement. You have put the attributive adjectives meum and dilectum in the nominative neuter singular, agreeing in case, gender and number with their associated noun animal, and the predicative noun avis in the nominative singular, agreeing in case (but not in gender or number) with the subject.
I don't know enough Latin to be able to tell you whether your sentence is an idiomatic way to say what your favorite animal is. I think it would be worthwhile to post another question about how best to express the idea "My favorite ... is..." in Latin.
As Asteroides points out, a predicative noun can freely differ in gender and your suggestion is correct. I want to add that it goes in fact further: in (very) rare cases a predicative adjective can have different gender too.
In the Aeneid (4.569–570) you can find the expression varium et mutabile semper femina. It is a complete sentence with an implicit est. The typical way to say that a woman is changing and fickle, you would say varia et mutabilis est femina with feminine adjectives. With the neuter the adjectives can be thought to be substantivized so that it means "woman is a changing and fickle thing".