Chris determined the cause of this issue. Unfortunately, it is not good news.
The cause of the problem is that the balances being displayed on the website do not include the transaction fees charged when customers click "ignore payout threshold." The 0.06 litecoins is equal to the transaction fees incurred over the lifetime of the account by ignoring the payout threshold. Therefore, these coins are not actually owed, and the payout script is functioning properly
We will fix the bug in the website as soon as possible so that the correct balances are displayed. Sorry for the confusion, and I wish that the answer was more positive.