How to add custom permission for a node?
A possible alternative is to use the Rules module, and create a rule that is a variation of the rule mentioned in my answer to the question aboutr "How to make the permissions of a module more granular?".

The variation in your case will be something like so:


Use the path specific to the node of your case.
Adapt the "Rules Action" to fit your needs.


You may also want to add some "Rules condition(s)" related to the "role" that is required to allow actions like edit, delete, etc. If you have multiple actions like that, you either build one such rule for each action, or you use the Conditional Rules module to make it all fit in a single rule. Obviously, this approach does not "introduce" a new permission that can be assigned by an admin (which is not a requirement either in the original version of this question, right?).