Immediate/continuous launch readiness is something of an essential feature in this role, so you don't compromise on it.

In general, solid rocket motors consist of particles of fuel and oxidizers suspended in some sort of plasticy/rubbery binder. These binders are usually some fun complex polymer. The Orion rocket motor apparently uses a form of polyurethane - which produces hydrogen cyanide gas on combustion.

Desirable properties of binders include: having a liquid form that can cure into a solid form in a controlled way (this makes manufacturing the suspension easier/possible), curing into something that is somewhat elastic/rubbery (I suspect easier handling, and more stress tolerant), and ideally also somewhat usable as a fuel (so that it's not a dead weight). I suspect that most polymers with those properties all have nasty combustion products one way or the other.

Some things to consider - most rockets, especially interceptor missiles spend most of their time NOT being fired. They are being stored and handled and tested, but not fired. That means that when considering your choices, you can't just blindly select for exhaust toxicity + rocket performance - you need to consider the operating cost and safety as well.

Its for this reason why solid motors are a go to - they are much easier and safer to handle than liquid fueled rockets throughout operation. They also have the desired properties in that they are effectively always ready to launch - you don't need to worry about managing fuel tanks or whatever.