Can a service member who arrives at a new assignment early apply for BAH before signing in to the reception battalion?

SSG Scheil you will get the previous station BAH until the date you sign in meaning when you process finance they ask for your DA-31 to see when you signed in. Let say you sign in Feb 1st you will start receiving BAH from that date onward. My recommendation would be to sign in and get you 10-day permissive TDY if the new location's BAH is higher than the other location you coming from.

No. You're not "at" the new assignment until you sign in at the new assignment. I've signed in and back out on leave before, but that takes some advanced coordination with your gaining unit. That approach might help serve the same purpose.

Why not get house hunting permissive TDY on your DA Form 31 from your losing unit before you leave? That is what it is designed for! Use caution some hard duty stations like Fort Drum, NY have specific local regulations forcing you to sign in first before housing will allow you to sign a lease. I learned that the hard way. That Command wants you to live where it is most beneficial to that Command for on call alert status. They will force your hand in a minute if you break local housing regs. Good luck in the house hunting.