Is it possible for water to ooze out through the ocean base and extinguish the fire at the Earth’s centre?

No it is not possible.
There is no fire at the center of the earth the center is most likely a molten liquid of nickel and iron. The core is hot for two main reasons, leftover heat from the friction created impact a gravitation when the earth formed and from the radioactive decay of heavy elements like uranium.
Because the earth is so large the amount of heat it contains is enormous, it’s so hot and so large that it takes heat a long time to make its way out from the center and into space
It is so much that it would take something like 90 billion years to cool off. For perspective, that is more than 10 times longer than it will take the sun to burn out and become a white dwarf star. That amount of heat could easily boil all of the water on earth many times over without even making a dent.
It seems like there is a lot of water in the oceans, but they are actually tiny compared to the planet’s mass.

As you can see here the oceans are really just a very thin film of water when compared to the entire earth.

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