Ok, found something that seems to work.
First, weigh this many coins from each stack:
Stack A B C D E F G H I J
Round 1: 3 3 3 3 2 2 1 1 0 0
There are two possible outcomes. I we're lucky,
the result is a multiple of 18 grams, all the weighed coins are genuine, and the fakes are in one of the remaining two stacks. Since we now know the weight of a genuine coin (result divided by 18), we can weigh one coin from stack I to identify which one is the fake stack.
If we weren't so lucky, we know that I and J stacks are genuine, and we'll next weigh these coins:
Stack A B C D E F G H I J
Round 1: 3 3 3 3 2 2 1 1 0 0
Round 2: 0 1 2 3 1 3 2 3 3 0
This should uniquely identify the fake stack:
Calculate the distance from the multiple of 18 that is nearest both cases.
Then compare the cases, and see what happened to the error:
* Vanished: A is fake
* went to one third: B is fake
* went to two thirds: C is fake
* no change: D is fake
* was halved: E is fake
* increased by 50%: F is fake
* doubled: G is fake
* tripled: H is fake
Finally, we need to rule out overlap, because we weighed fewer than 22 coins. (The maximum possible error is 10.8g, so the nearest multiple of 22 would have been uniquely defined.) So we need to figure out if there are any particular numerical combinations that would allow misinterpretation.
The cases that would be easiest to confuse with each other are "went to two thirds" and "increased by 50%". (Sorry for pulling this out of a hat, don't have the time for a rigorous argument now.) If the F coins were exactly 3.6 grams too light, we'd get a "10.8 grams to 7.2 grams below a multiple of 18", which could also be interpreted as the "7.2 grams to 10.8 grams above a multiple of 18" which would mean that C coins were exactly 3.6 grams too heavy.
Since the fakes are guaranteed to be less than 3.6 grams off the regular weight, this is not a problem. (TODO: rigorously check that the rest of the cases would require an even bigger weight difference for the fakes. EDIT: maybe tossing in the J coins for the second weighing would help, too.)