How can we get our Live Release Rate without using Evaluation Categories?

Try report 344 "Asilomar Accords Report [Uses Euthanasia Reason]" (Reports > Shelter Reports)

This report side-steps the possibility of user error by allowing the site to set up evaluations against Euthanasia reasons. This can be done via Admin > Edit Drop Down Lists > Euthanasia Reasons / Animal Features > Maintain. 

Simply click Edit against a euthanasia reason and then assign an Asilomar category of Unhealthy Untreatable, Treatable Rehabilitatable, Treatable Manageable or Healthy and from then on the report will use this to determine these things. 

Even if no Asilomar category is assigned, it will still calculate the Live Release Rate.