Any ideas where to start?
Have you thought about contacting the company?
Call them up, tell them what you are wanting to do (have the model number of the device handy), be professional and tell them how you want to integrate the Arduino to control the system. DO NOT tell them that the remote was less than desirable - you want them to understand that you are wanting to augment the system.
You might also inquire about obtaining another remote. Make sure in your dealings that you mention you will pay for the things you are requesting. This lets them know you are serious. In some cases, by doing these things and acting/speaking right, you can open doors and obtain product generally for free. Ask about specs and service schematics or service manuals. They may have some lying around (especially if it is an obsolete product).
I recently did this with a pair of bench power supplies that were at least 15 years or more old, that I bought from a local electronics junk/surplus yard here in Phoenix, Arizona (Apache Reclamation and Electronics). I first emailed the company, then followed up with a phone call. Not only did I get actual originals of the user and service manuals, but they shipped them to me on their dime; I didn't have to pay a nickel.
Don't get me wrong; it doesn't work like this with every company, but generally, if you ask nicely, you would be amazed the information and help companies will give for their products, even junked or obsolete ones (which can sometimes be like pulling teeth, unfortunately).