I would like to be able to use a variable inside the like_regex predicate, at least for the pattern.
You can use a format function to build it dynamically. Unfortunately it is a bit of a pain since you need to do escaping; which is a pain for regex. SQL scope doesn't have this problem so moving your logic outside of a json is should seriously be considered before trying to construct dynamic jsonpath expressions.
I get the impression we are conforming to a standard here so even proposing a patch to change this behavior would require some convincing to deviate from the standard on this point. Though I could see adding a new format escape and related quote_jsonpathliteral function to be something we'd be more open to in order to make dynamic json path expressions more easily doable.