That's great advice to break it down into smaller jobs. I clean my drums by removing all the lugs/HW, using a soapy washcloth, like you are washing your face before bedtime:), rinse with another one that has no soap, then dry it with a dry towel.....then it seems inevitable that some water was soaked into the shell (wood) at the holes. I then simply take my (hair) blow dryer to it and it's done. Yeah, don't want long-term moisture/water on wood shells! NOTE: use SOFT towel that will not scratch, like a microfiber towels that are made to use on cars.
The probably issue was fairly confidently concluded as too much soap (too strong a concentration of the Dawn) and too long of soaking. So, much learned by that experience I am sure (myself included).
BTW, that white build-up on the inside of the lug is just the bare metal reacting with the soap....no big deal, brush it out and move on. My experience has shown that that issue is much worse when Bon Ami is added to the solution, so I just recommend(ed) straight water and Dawn.
BTW, Matthias, I think several of us are getting mixed up on what all drums you are cleaning, dealing with. Guess it is not a big deal, as we are dealing with problems encountered, not drums persay (doesn't matter to keep it all straight). It is good to know that you can work on these kits all you want considering you have another kit (your red Pearl Export) to keep you 'up and running'.
I remember one time I feared these chrome problems and did not have the time to do any cleaning before bed one night, so I emptied most of the water out of the (plastic large) tub (containing my soaking parts) and then added the water back, resulting in significantly weakening the soap concentration. It may have made a difference and I may have averted this peeling chrome problem.
BTW, a little 'trick' so to speak. Stay in touch with other drummers/friends. When somebody is planning to buy something online, ask to get in on the order and you can save alot of money on shipping. Example is at drumfactorydirect they offer free shipping if the order is $100 or more, let your friend buy something to add to the order, bumping it over the 100 mark, and both of you are getting free shipping. You get the idea.
Happy drumming.........
John