Hello,
I'm a bit surprised that your characters are aware of the maid's condition at all - could she afford to see a doctor? - but there are probably reasons for that. I don't think that there is one surefire thing to cause death of a woman with eclampsia or pre-eclampsia. It would have to be a combination of factors, probably: bad food, a lot of stress, a lot of work, not enough sleep, irregular hours. All of which would apply to your unfortunate maid character. The antagonist could simply work her to death, make her get up when she's been resting or eating and ran around on some foolish errand, rinse repeat. Add to that a few traditional punishments, like kneeling on gravel or on chains with a stone on her head, and it won't end well for the maid.
If the antagonist is pressed for time, maybe poisoning or pushing the maid down the stairs?
Also, I second frimble's words about a death of a brothel maid being of practically no consequence and very easily covered up. Actually, my own main character is a servant in a Tokyo brothel in the 1880s, and he's well aware that any misstep on his part might be fatal, because no one cares if he lives or dies. The brothel owners have the police AND the local gangs on their side or simply in their pocket. I would imagine it is the same for your time and setting, even more so, given that it was a politically unstable time for China - the Qing dynasty had just come to an end, the Kuomintang and the government of Yuan Shikai were at daggers drawn if not outright fighting (depending on the time of the year), and a lot was going on specifically in Shanghai.