I recommend to not implement the test suite in Ranorex again. The simple reason: from a watcher perspective, both automation tools will perform the same actions, if everything goes well.
Instead, make a qualitative analysis of Ranorex' pros and cons compared to TestComplete. A search for "TestComplete vs Ranorex" will probably bring up some search results. Then, try Ranorex yourself if you haven't yet and compare the capabilities to TestComplete. Most probably, you will not have to implement a full test suite to get a comprehensive picture. Instead, experiment with tiny use cases.
Before doing this, get yourself clear about your customer's needs. Will he be maintaining the test suite himself later? What are the testers' skill levels? Budget constraints? What are his concerns regarding TestComplete, why look into Ranorex at all? Is TestComplete the currently chosen technology?
A decision like TestComplete vs Ranorex is also a strategic one: you cannot clearly anticipate the effects; you will make a guess what will be best for the future.
If you are no advocate of Ranorex yet (since you recommended TestComplete to your customer in the first place), be open to have your opinion changed. I myself have done an analysis in the past where we evaluated a transition from TestComplete to Ranorex. Read Ranorex' User Guide and the API documentation. In my opinion, judging based on these two and comparing it to TestComplete's equivalents, Ranorex is the clear winner. But the biggest advantage in my opinion: Ranorex builds everything to native C# code, whereas TestComplete is powered by some script language in different dialects. With some imagination, this implies huge benefits.
If you fear to lose your customer because you cannot support him with a pro-Ranorex decision, I am afraid to see no other option but to disguise your customer.
Full disclosure: I am not associate with Ranorex. Just a confident user.