Is it permissible to work a second, illicit job, knowing that the social welfare [department] will deduct from my salary if it finds out?

The salary you receive from social welfare is a conditional gift, and it is not permissible to take it unless the condition is met, based on the Prophet's (PBUH) saying: "Muslims are bound by their conditions in what is lawful." Therefore, receiving funds from social welfare while simultaneously engaging in another job is not permissible, as it constitutes a breach of conditions. If the other job is also unlawful in Sharia (definition), then you have committed another sin. And if it is prohibited by the laws of the country, then you incur sin by engaging in it because you would thereby be breaking a covenant.