There are no sure-fire ways, but in the time since I first asked this, I have managed to get bronze working first a few times. The trick is research pumping hard early, which is easier in a Japanese civ -- if you are in an island world.
My question, however, was incorrect about its assumptions. Unlike most other purely military techs, being first to bronze working gets you nothing. Unlike practically every other tech in the game, being first to bronze working doesn't even seem to notify you at all: I only discovered I'd been first to it by going into the tech tree, and then verifying in Civilopedia. No pop up appeared for me when I've managed to do it.
Thus, based on my findings, I've now switched my strategies a bit: there isn't much incentive to hard-rush bronze first. It's a vital tech, no doubt, but the premise of "get it first for the free unit" doesn't apply. So go for it early, but don't kill yourself getting there.