Cassell's Dictionary of Slang doesn't have an entry for "brick sandwich," but Irving Allen's The City in Slang, a slang history of New York City, does have a short entry on Page 168:

"One type of free lunch was more apparent than real. In 1896, New York State enacted the 'Raines Law' to tax and control the retail liquor trade. One provision prohibited saloons from selling liquor on Sunday, though hotels could sell liquor with meals. Many saloons quickly became bogus hotels and began to serve liquor with equally bogus meals. Hence the birth of the so-called Raines Law sandwich or, in slang, the brick sandwich.

"It was two pieces of bread with, it almost seemed, a brick between, though this concoction should be taken more figuratively than factually. This inedible meal was set out on a table for nominal compliance with the Raines rule. The brick sandwich usually had real filler, but the customer was not supposed to eat it, and contemporaries described it as so repulsive that no one would want to, though a few dared. Jacob Riis in The Battle with the Slum tells a story, reported in the New York Herald on April 8, 1902, of the police who broke up a fight between a Brooklyn bartender and a hungry customer who had tried to eat the brick sandwich. The police restored the stage-prop sandwich to the bartender and made no arrests."

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I find that unsatisfying in light of the context, so what I suspect it may refer to is the practice of carrying bricks in lunchboxes or carrying bricks with their lunches. It isn't unprecedented -- at least in labor history -- for workers to carry bricks with them as a means of defense or agitation, and a lunchbox or purse laden with a brick is a pretty hefty blunt weapon when swung. Imagine that one scene in Full Metal Jacket, but with bricks instead of bars of soap. Even thrown, a brick is a pretty powerful weapon, but there aren't bricks around all the time -- hence the need to carry them. (Cassell's has the wonderful definition of an alley apple being a thrown brick in the 1910s.)

Interesting question.

I checked the OED. Nothing for "brick sandwiches," or even any entries with the words brick and sandwich in common. Since the context is violent, I thought maybe the phrase was akin to the idea of a "knuckle sandwich"--but that idea seems unlikely because the OED dates the phrase knuckle sandwich to 1973.

I also searched for knuckle sandwich in a couple of databases of archived newspapers and couldn't find the phrase any earlier than the late 1940s. I couldn't find "brick sandwich" in the right context at all.

There was apparently something called a "brick sandwich" advertised in the newspapers (mostly 1920s--one 1908 hit and a couple 1930s hits) and a "gold brick sandwich" (1950s) that were actual sandwiches for sale in cafés or lunch counters, but that doesn't seem related at all to the brick sandwich you're looking for. I think the term was describing new-fangled processed lunch meat or cheese that was cut from a brick shape because I also saw grocery advertisements from the same time period for "brick sandwich meat" or "brick cheese."

Not really an answer, I know. I've just sorta given what little evidence I could dig up of what it isn't.

As others have pointed out, from the context, a "brick sandwich" seems to have been local slang for something the women carried either to protect themselves or to have in readiness to throw at the cops.