How should I map an object that applies in all test environments

That specific object may not be found but TestComplete uses the parent objects as well to find the object... so, it could be further up the tree of ancestor objects as to where the problem may be.  In your screenshot, your object is a child object of a web page...  since you're talking about different environments, I'm assuming the web page is bening mapped by a URL that is different in each environment.  What you should do is, on the page object, wildcard with the asterisk (*) character the leading parts of the URL that are different between environments.