Is it permissible to compromise on obligations, such as shaking hands with a non-mahram woman for the purpose of da'wah, and what are the controls for such a compromise?

This issue is "Fiqh of Prioritization" or "Consideration of Benefits and Harms," which is a foundational principle upon which Sharia was established. It aims to achieve and perfect benefits, and to disable and reduce harms, while prioritizing the better of two good things and repelling the worse of two evils. This prioritization applies to necessities (daruriyyat), needs (hajiyyat), and embellishments (tahsiniyyat).

Prioritization is divided into: 1. Conflict between two good things: The better of the two is prioritized. 2. Conflict between two bad things: The worse of the two is averted by enduring the lesser. 3. Conflict between a good thing and a bad thing: The more preponderant is chosen, whether it be the benefit of the good thing or the harm of the bad thing.

Thus, a bad thing may be tolerated to avert something worse, or to achieve something more beneficial than abandoning it. And a good thing may be abandoned if it causes the loss of something better, or if it entails a bad consequence whose harm outweighs its benefit.

Regarding shaking hands with a non-mahram woman, the harm of aversion can be averted by explaining the religious ruling and honoring the relative without shaking hands. Prohibited acts are of varying degrees; what is prohibited as a means (like shaking hands) may be permitted out of necessity or for a preponderant benefit. Prioritization is not limited to recommended (mustahabbat) and disliked (makruhat) acts but encompasses all categories.

This is a delicate matter that requires deep understanding of benefits and harms and their respective degrees.