Is it permissible to devise a stratagem for a sick sister so that she may be treated, or to advise her that not treating her could lead to dire consequences, given that her husband refuses her treatment and prevents her family from doing so?

The husband is obligated to provide medical treatment for his wife, and this falls under her maintenance. If the husband refuses, the wife may treat herself at her own expense or at the expense of her family. She is permitted to leave without his permission to alleviate harm from herself.

As for the stratagem of lying, it is permissible out of necessity to enable her to obtain her right. Speech is a means to achieve objectives. If a commendable objective cannot be achieved except through lying, then lying is permissible. If the achievement of that objective is permissible, then lying is permissible; and if it is obligatory, then lying is obligatory. It is recommended to use equivocation whenever possible, but if it is not possible and direct lying is used, it is not forbidden.