In other words, I could now see the theoretical possibility of a just execution, (however remote that may be) but I cannot conceive of any government that has ever sat in power anywhere that deserved the publics trust in such a decision, in that regard all fall short in my opinion.
By this logic we should not trust any government to make decisions regarding war either.
Saint Paul recognized the right of pagan, Neronian Rome to execute criminals (see Romans 13); it’s silly of you to assume that the Church has just been plain wrong over 2000 years for acknnowledging the rights of Christian nations to do the same. Fallible humans really do have this right and duty.
The second part of my question is linked to the first: disagreement with the Catholic church in a matter of faith or morals could be sufficient reason for the state to execute an individual? In other words, are you stating that the individual who disagrees with Catholic doctrine may be considered in some way a public danger of sufficient criminal magnitude to merit execution?
Simply being a (private) heretic does not constitute a grave public danger. But in a Catholic state, a public heretic who leads so many souls to hell and works to subvert the Catholic faith does constitute such a threat, and the State should employ measures against him. I would argue that execution should be a last resort, only after other punishments have been exhausted (i.e. fines, imprisonment, exile, etc.).
People have a right to believe what they want. But in a Catholic state, they don’t have a right to publicly express and spread beliefs which endanger the salvation of souls and impede the mission of Christ’s Church.