How to resolve mixing singular and plural on the same noun

The problem is with your chosen pronouns. Observe how smoothly it flows once a few corrections have been made:

China has as much claim to North Korea as it does over Tibet. That is, if it does decide to invade and occupy North Korea.

Why did I choose the pronoun it? Quite simply because you said China has. You should maintain consistency among your pronouns, so if you use the singular "has", you are treating the noun (China) as a single, collective unit. That is, you're talking about China as the country in general as a single entity.

If you'd prefer to use the plural pronoun they, then the change is as simple as replacing "has" with "have" and "China" (singular) with "The Chinese" (plural):

The Chinese have as much claim to North Korea as they do over Tibet. That is, if they do decide to invade and occupy North Korea.

China has as much claim to North Korea as they have over Tibet.

I don't actually agree that the use of verbs with different number agreement to describe the same logical entity in this sentence is a problem in terms of grammar. (Some people might object to it as a matter of style.) It's usual for verbs to agree with their immediate subjects, not necessarily the antecedent to their subjects, and just because an expression refers back to a preceding noun phrase doesn't mean it has to match in the agreement categories of number and person.

Compare:

Your Highness is as generous as you are eloquent.

"The Merchant's Talisman", by Paul de Musset (in French), translated by G.J. Knox, The Literary aspirant magazine (Jan 1846) p. 171

Other examples of this kind of mismatch may arise with the perhaps badly named "singular they" as in:

any student has as much time as they want to take to complete the test.

Kill the Messenger: The War on Standardized Testing, by Richard P. Phelps

Other relevant posts:

• subject-verb agreement for singular they

• Why isn’t singular ‘they’ used with 3Sg verb forms?

• Why do we say "You are" when you is singular instead of "You is"?

• Historical usage of "was"/"were" with "you"