Up until right now, after looking up the word "foreigner" in a few dictionaries and finding nothing relevant, I would never have guessed that it is British slang. I have always assumed that the word foreigner was a homonym, since it's use in this context is quite universally understood by people in the UK.
The most relevant "source" (and I use that term very loosely) that I could find was Urban Dictionary, which defines it as:
Common British slang for unofficial work, often making use of company equipment/ time/ premises but solely for personal gain, unrelated to the interests of the company [..] but DOES NOT hold racist connotations in this context. With notable exceptions, doing Foreigners is a fairly accepted part of British working life and, depending on the workplace, can even be done with managemental consent.
I'm not coming to the pub this lunchtime, I'm staying at work to do a Foreigner for a mate of mine.
Wiktionary's definition isn't quite as clear (unless this is a different meaning again):
A private job run by an employee at a trade factory rather than going through the business.
I also found a thread on Word Reference asking the same question.
According to this webpage, the phrase "doing a foreigner" may originate from the 18th century practice of Liverpool ship owners registering their ships abroad and carrying slaves for other countries, in order to flout the ban on the slave trade; although this isn't verifiable.
What actually prompted me to wonder if there is another way of saying "I still need to do that foreigner" is that I said it recently to a colleague in a restaurant, but it was overheard by a passing non-British mixed-race waitress, who - judging by the shocked look on her face - clearly misunderstood the meaning.
What is a less ambiguous way of saying this? If it does originate from the slave trade, then it probably isn't politically correct to say in any context; despite its widespread use.