Recommended starter switch for 582

If you are running a 582 it is probably because of weight considerations. Stay light and simple, use 2 toggle switches. This is by far the most dependable system. 

I agree with your reply, but my question is about the switch to engage the starter relay. 

A push button, normally open switch rated to supply the solenoid and configured for your application will do the job. 

Are there any specific switch manufacturers who can be trusted for quality and reliability?

Any switch that costs more than $2 will fit the bill.

It only delivers power to a solenoid, Not he actual starter so it only switches an amp or two and usually only once per flight.

There is no need to overthink this one.

If it looks like a quality switch it probably is.