That's assuming the Chinese even wants or needs an oversea military base. Any military needs overseas can, and should, be done under the auspices of the United Nations.
China should work on the principle here, the more friends you make, the less they become your future enemy. This would help your trade, boost your soft influence and promote your values. Not going around taking the big stick (military action) and whacking everybody.
China has perfect rationale for a base in Tanzania or Seychelles, because she's providing global public goods against pirates in Gulf of Aden and maybe even in West Coast Africa too.
Better yet build an artificial island base instead in international waters.
That will open such a can of worms in real life, though it is an awesome James Bond-ish idea.
A military base in the Western Hemisphere, even an elementary one, would provoke the US more than China wants to right now. I still think the first Chinese military overseas base will be a duel use Navy/civilian installation in either Tanzania or Seychelles, because anywhere near Pakistan or Sri Lanka would be too provocative to India.
The Russians would probably do so before China as they've had bases in Cuba and such in the past, so for them it's simpler.
That's a legacy from the Soviet Union, and not something China had anything to do with. But it would be provocative for China to try for a base anywhere in the Western Hemisphere, plus there's nothing in it for China anyway. I mean why piss off the US when there's nothing to be gained? Xi Jinping isn't Vladimir Putin.
In the lower Atlantic Ocean between South America and Africa are a vital interests to China and more so in the future.