Chinese monarchism was functionally dead by 1917, as evidenced by the collapse of Yuan Shikai's empire and the completely abortive attempt at a Qing restoration. The Mandate of Heaven is also something that's been hugely misunderstood and exaggerated in Western sources- yes, it's a way of explaining the rise and fall of governments but plenty of Chinese regimes have endured crises without anyone declaring 'They've lost the Mandate!'
So claiming the Mandate wouldn't do anything to increase Chinese support for the invasion. In fact, it actually would weaken the Japanese:
1. It undercuts the legitimacy of the puppet Manchurian regime even more, though this is at best a minor point given that no one took Pu Yi seriously anyway.
2. It makes it much harder for people like Wang Jingwei to defect. Wang apparently seems to have genuinely believed that the war was lost and collaboration was the best way to save what could be saved- but that's contingent on some kind of nominally independent Chinese regime. If the Japanese have claimed the Mandate, they are in fact declaring their intention to rule China directly. This will go against their stated propaganda in OTL, and will encourage even more resistance. The stakes have been raised- the chance for some kind of peaceful, if puppeted China has gone.
3. It's going to seriously hurt the Japanese diplomatically. They had enormous difficulty in our timeline trying to convince anyone- anyone bar the Nazis - that they had legitimate aims in China. Their hopes in the early years seem to have rested on forcing the Chinese to recognise Japanese influence north of the Great Wall (their ambitions snowballed, obviously, even as their armies became bogged down) and hoping that the other Great Powers would recognise the end of the Open Door as a fait accompli. Here, they are explicitly stating that they are going to exert control of the whole of China- this makes clear that they are going against the interests of the Soviets, the British, the Americans and the French. Even worse, they're coaching it in exactly the kind of terms that will allow China's lobbyists overseas to paint this as a contest between a beleaguered, modernising, Republican China and a 'fanatical, Emperor-worshipping' Japan on the other.
In summary, Tokyo claiming the Mandate would be like Britain announcing on D-Day that the French resistance should welcome British troops because George VI is the rightful King of France and the Angevin Empire.
It would... not be received well.