Completing Godzilla: The Game story mode with Hard difficulty is not painful if you use the right tactics and make your shifts with strategy.Here is the complete Guide on how to beat the stages in hard mode.
Improving your Health and defense is must since the final three stages are tougher, and so you need to be upgraded/level up in evolution mode. This will help you to sustain more attacks, and also you will be able to take more hits from the other enemies.
For the Combos and Fury counter you need to focus on it as much as you can. You can keep on improving your combos by destroying the buildings in each and every stages. To match the beast/kaiju in the later stage you must grow faster by increasing the multiplier. Also, increasing the fury counter is a must to defeat the beast.
If still end up being smaller than the kaiju just don't panic and focus on the Shield defense to survive. You can also use a lot of finger missile attacks to give him more damage. Some how you will make it. And if you have grown bigger than the kaiju then it will be very easy to defeat him, but you need to take the right attack and defense as well. Best of Luck!!