There are liquor stores in Manama. The signs say they will only sell to non-Muslims, so it's best to get some Anglo or Indian friends.
I'm sure there are additional outlets; these are the ones I know about first hand.
BMMI and Gulf Cellars are just south of the Pearl roundabout. These were hard to find if you don't know exactly where you're going. As you go around the roundabout, you'll see a car wash with a big sign, "Modern Car Wash". The stores are directly behind the car wash. Exit the roundabout after you pass the car wash sign, in front of the Marina Mall. Make the first right, into a parking lot between a petrol station and a market. Go through the market parking lot. At the far end of the market, turn right. Go to the end of the road, turn left to go around the Modern Car Wash. Turn left again, and BMMI is immediately to your left. There's a white sign with red letters. The two Ms are sort of superimposed on each other, so at first glance it looks like BMI. Gulf Cellars is almost directly next door as you continue down the alley toward the police station (on your right).
Gulf Cellars also lists locations at the Gulf Hotel (across from the Grand Mosque and the new palace) and the Palace Inn Hotel (same street as the Gulf, but farther south). These hotels are marked on every Manama map I've seen.
The liquor stores are closed on Friday.
But save yourself a few dinars and take advantage of the duty-free shop that is immediately after passport control in the airport. Nice selection, and it is cheaper. You can bring in one litre of liquor and a six-pack of beer.