Russian influence and Chinese transit

Would Russia allow China to transport troops through territory under Russian influence?

Yes. It remained Chinese territory, and Chinese resistance to Japan could serve Russian interests.

The region remained legally Chinese despite strong Russian influence, so troop movement did not require Russia to surrender its own sovereign territory. Influence and ownership were not the same thing. Russia also had a strategic reason to tolerate Chinese movement because stronger resistance could limit Japanese expansion. A Chinese defense of Manchuria would therefore preserve or enlarge Russia's relative influence rather than threaten it immediately.