Adding a quick note in case this helps anyone. It seems like Hue migration process will try to make the new bridge to use the old bridge's IP. That's why it's important to reset the old bridge so it will release the IP.
I previously set static IP for my old bridge on my router, so the new bridge wasn't able to request the old bridge's IP, which caused the Integration to fail to initiate. Manually fixing the IP assignment fixed it for me.