No other parts that I can think of. Just filter and gaskets. It doesn't hurt to look at hoses and stuff while you're under there...y'know, check the condition. Also a good time to inspect driveshaft and other components.
I would have thought that there is a drain plug on your tranny pan??? There is one on my 1990. Maybe they changed it in later years. I know they went to a "sealed for life" transmission but I didn't think that happened until 1995. Anyhow, that's side points...
I'm not sure on the 3HP22 parts. As far as I know (and I have been known to be wrong at times), all E32 BMW's (7-series built between 88-94) have the 4HP22 or 4HP24 for automatic transmissions. You can get the proper part numbers by going to
http://www.bmwfans.info, input the last seven digits of your VIN (I highly recommend you memorize those last seven digits or write them down for easy reference), and you can even cross reference part numbers by clicking the part number. I have found so many cheaper parts because some parts for my E32 750iL were also used for a E30 325i. Any how. That's a good link to favorite under a BMW folder.
As for cleaner, I believe there is a transmission flush agent you can add. This does require that you drive the car however. Most often, just changing the fluid is all that is required. The residual that is left behind from the old fluid is not enough to have an impact on the new stuff.
The TRANS PROGRAM error can occur for any reason, but it isn't coincidence that you experience this problem and saw this error message. Sometimes it's a loss of pressure in the valve bodies and the transmission doesn't shift properly, doesn't shift at all, comes out of gear or gears don't engage. This message can appear to electrical problems and sensors. Cleaning the gear lever selector sensor can fix this and cleaning the A/M switch next to the gear lever (or S/E/M switch if so equipped). The output shaft sensor can cause this, down to a duff clutch pack in the torque converter. Hopefully it's a problem with the former as the torque converter can get fairly expensive if you don't have a trusted mechanic on hand.
Basically, the TRANS PROGRAM has put the car into what is known as LIMP MODE. If you sit and think about what LIMP MODE might mean, it's exactly what it implies. Your car is "injured" in a sense so it is limping. Your car will derate power and lock your tranny into high gear so as to reduce the risk of causing further complications. The TRANS PROGRAM is also why you are not seeing the E or the M next to the selected gear display. This was back in the 80's when BMW thought of this, almost 30 years ago...yeah, way ahead of it's time!!! Most of the technology that our cars had back then has only begun to be installed within the last 10 years on most "low end" cars.
As a final note, one web site I order parts from is
http://www.autohausaz.com. They are based out of Arizona and are fairly cheap, even with shipping to Canada. Check there with the part numbers you have cross referenced from bmwfans.info.