As an overall trend, people act in their self-interest. At best people act in their long-term self-interest. So if you want to convince people of something, appeal to their self-interest. This may need to be an indirect appeal.

Many of the processes we pejoratively call "cognitive biases" are actually true. Either in the sense of being useful heuristics for everyday circumstances, or in the sense of just being generally true (ie the prototypical grandma being right and a PhD scientist being wrong).

For example, hyperbolic discounting is completely rational in the face of uncertain risks. This is clearly the case when planning for the far future. While one might care about future beings in an abstract sense, it doesn't make sense to include their well-being in ones decision making as it has been discounted to approximately zero. As an extreme example: I fully agree that humans outside my light-cone have the same moral worth as those inside my light-cone, but since I can never effect those outside my light-cone (assuming they exist, which is not something we will ever know) I don't factor them into moral decisions.

I'm not sure that internal consistency should be the highest priority. If it is, that implies constraints on the applicability of a moral theory (ie some questions will be undecidable). Which may be fine, just be aware of that tradeoff.

Impossibility theorems are pretty common in mathematics. Arrow's Impossibility Theorem will apply to many ethical frameworks. Where it doesn't, other impossibility theorems (Godel, Gibbard, Holmstrom) are likely to apply. If nothing else, reading up on this class of theorems may be interesting.