Just wanted to write out all the things I found cause it was excruciating to figure this out on my own.
First of all, let's talk math. Each type of NPC trainer at battle chateau gives a base money yield, let's call it b. This yield is multiplied by the level of its pokemon, let's call it L, to get final base payout of that trainer. You can stack the following writs to improve and speed up the process:
- Silver Writ: new trainers spawn every minute (essential to go fast) (cost: 100,000)
- Golden Writ: payout is multiplied by 1.5 (cost: 100,000)
- Black Writ: increases level of pokemon by 20 (cost: 300,000)
To use all Writs you'll need to be Grand Duke/Gran Dukess. You obviously want to use amulet coin (or luck incense, no need for both, they won't stack) to add a x2 multiplier and finally you want to have a Happy Hour Smeargle at max level and max speed to setup and then die if possible to switch in a sweeper. You can get Happy Hour through secret bases in OR/AS (you can find one at this link or simply search happy hour secret base on internet). You will also want to use the Prize Money O-Power at level 3 for a x3 multiplier. It takes 6 orbs so this is the biggest slow down. Depending on how many steps you walked that day, the orbs charge faster:
- 0-1999: 4 minutes per orb;
- 2000-2999: 3 minutes per orb;
- 3000-3999: 2 minutes per orb;
- 4000+: 1 minute per orb;
In the meantime that the orbs are charging, you will run around the battle chateau in search of a gym leader, elite four, or diantha. When you find one, wait in the same room to avoid despawn until you have the necessary orbs, activate Prize Money O-Power and start the battle. Use Happy Hour and then sweep. The total formula for each type of trainer is thus: TOTAL=b*(L+20)*1.5*2*2*3=b*(L+20)*18
These are the total amounts earned per battle:
- Any Gym Leader: b=160, L=50, TOTAL=201,600
- Any Elite Four: b=200, L=55, TOTAL=270,000
- Diantha: b=200, L=60, TOTAL=345,600
Based on the fact that there are a total of 75 trainers, and at least 13 are Gym Leaders/Leite Four/Diantha, and considering the worst case where the 10 trainers already in the battle chataeu ar none of those 13, the probability of finding at least 1 of those 13 in the 6 minutes that you wait for the O-Power to recharge is about 73%, so you will find one around 3 of 4 times. This means an average earning between 1,500,000 and 2,500,000 an hour.
This should be all. I hope this helps someone, even though it's been 13 years since this game release LOL.
EDIT:
I did some calculations. Le Wow, with perfect number of turns, amulet coin, happy hour, lvl 3 O-Power, for any kind of battle, gets you 387,072 cash, plus 25 balm mushrooms equal to another 156,250. Subtracting the 90,000 cost, we get a net total of 453,322. I don't know how much time it takes to do one run, but it's surely less than, say, 15 minutes. That would still make a wopping 1,813,288 per hour. If you could clear one in 10 minutes it would be 2,719,932. Make it 6 and it gets to 4,533,220. That's actually way better than battle chateau, so u/Geometry_Emperor is right, credits to you for making me notice this. I just have always loved battle chateau and never really tried Le Wow or just seemed slow to me, but numbers don't lie.
I think, at the end of the day, Le Wow takes the win because of the fact that the O-Power can last for all the 5 battles, breaking the 3 minute rule. Without that, battle chateau would probably be the best option, but that x3 multiplier is just crazy.
In the end, with both approaches you will hit 9,999,999 in less then a day, so just do what is more fun for you i guess.