Why would a bookie refuse a bet when he can back it back at a bigger betfair price

Because quite often he cant. Betfair markets move quicker than the bookies are able to update their prices. Plenty of arbs available everyday on the horses.

And if the horse wins or loses he will still win.

he can tell the customer to hold on for a few seconds and tell him he has the bet after he balances his books/

but then he would have to offer the customer a shorter price.

THEY ALWAYS DO.

If you go into the city of Dublin and ask for 10 grand on a 1/1 fav THE BOOKIE will not bat an eyelid and take the bet after he has backed it back at 11/8.It happens every day.