There have been many attempts to change tax laws in non-Muslim majority countries like the UK, France and South Korea and in some (UK and likely France soon) these have been successful. But, South Korea's parliament again failed to pass the law with opposition from some Christian lawmakers who fear that recognizing the Islamic restrictions on interest would be against the country's constitution. I don't know enough (anything) about the South Korean constitution to know whether that is the case, but it is clearly at least to some degree based on a general Islamophobia because opponents also expressed fear that the Islamic financial products could result in funds being funnelled to terrorist groups.
Despite the fact that there is no basis for the belief that Islamic finance supports terrorism (Islamic banks and financial institutions are subject to the same restrictions on funding terrorism and being involved with money laundering), it does highlight the difficulty of expanding Islamic finance beyond the Muslim market. Similar baseless fears have led to laws banning "Shari'ah" in Oklahoma, which Haider Ala Hamoudi tried to understand on his blog Islamic Law in Our Times.
Some have called for Islamic finance to drop the 'Islamic' part of its name and the use of Arabic terms (which I have criticized on my blog). This would not do anything to defuse those who believe Islamic finance is somehow a nerfarious attempt to "impose Shari'ah law" in the West. It is likely that there is nothing that can be done to convince these critics that Islamic finance is either benign or beneficial. However, these instances where the general public on its own or through politicians take a broad brush to anything 'Islamic' suggest that there is not wide enough understanding about Islam in general or Islamic finance in particular.
I don't have a solution to approach the anti-Islamic finance crowd, but I think that there should be more of an effort by the Islamic finance industry, particularly from institutions outside of Muslim majority countries to become more active in explaining Islamic finance to the general public. Of course, these institutions are more than willing to explain to potential customers how Islamic finance works, but I think that is not enough. The industry needs to have some broader outreach to ensure that the dialogue is not defined by the opponents of Islam in general or Islamic finance in particular.
A cursory Google search turns up no Islamic finance advocacy groups. With all of the resources being applied to developing Islamic finance, particularly outside of the traditional regions where it has thrived like the GCC and Malaysia, it seems problematic that there is no group out there committed to explaining Islamic finance to the public. There are many spokespeople who can articulate the reasons why Islamic finance exists, how it works and why it is not the threat that some people claim. However, most of these people simply do not have the time to make a concerted effort to explain it to a wider audience because they are involved in tackling the most pressing issues in Islamic finance. If Islamic financial institutions truly want to broaden their market beyond Muslims, this should be one of the first things to set up. However, the benefits will accrue to the industry as a whole, so it is in no one institution's benefits to commit the resources to this important task. Therefore it is imperative for Islamic financial institutions to work together through an independent group to coordinate activities.