Struggling with handing an object if it exists, and ignoring it if it doesn't

You definitely need to use a WaitNNN method, but I think you need to try WaitNamedChild or WaitAliasChild since you are using name mapping.

You cannot use mapped names with WaitElement bc the object selector needs to be an XPath expression or a CSS selector. And the Find method cannot find an object that does not exists, so you'll have to use a Wait method in combination with the Exists property. Try:

def WaitAliasChildExample():
  # Specifies the Alias object
  AliasObj = Aliases.browser.pageName.formAspnetform
  # Checks whether the submit button has appeared within 10 seconds
  if (AliasObj.WaitAliasChild("submitbuttonDelete", 10000).Exists): 
    Log.Message("The submit button exists")
  else:
    Log.Error("The submit button does not exist")

[TestComplete reference].