Actually, NT$200,000 is not such a big demand as the starting-point for negotiation.
If I had been knocked down and “run over” by someone driving an extremely expensive car, suffered ligament damage and had to be put in a brace for six weeks, I’d consider such a sum barely sufficient to compensate me for the trauma, pain and suffering, inconvenience, loss of mobility, time wasted in obtaining medical treatment, curtailment of my ability to exercise, disruption of my sex life, difficulty in sleeping at night, worry about possible long-term effects of the injury, and so on. (I wonder what you mean by “run over”? It sounds rather serious.)
Perhaps the guy will be unable to work and will lose substantial income as a result of his injury/injuries. If so, you’ll be liable to compensate him for such loss of earnings, which could easily amount to a six-figure sum if, for example, he’s a skilled worker with steady employment.
If the matter ends up going to court, you’re all but certain to have judgment entered against you, regardless of any prejudice that the judge may nurse toward you because you’re a foreigner. In Taiwan, the larger vehicle is almost always held liable for a collision with a smaller one, and the policeman who attended the accident has already pronounced you to blame for it. As the driver of a car that in this country is considered a plaything for the very wealthy, you’d hardly be likely to gain much sympathy from the bench, whatever you might have to say about it. The judge would most probably be disposed to push up the plaintiff’s damages to the highest amount that could possibly be justified, and you’d also have to foot the bill for both sides’ legal costs. It could all amount to even more than the NT$200,000 that the victim has asked for.
If you negotiate with the victim, show a good attitude, express sincere remorse and sympathy, and plead your case well, you shouldn’t have too much difficulty getting him to settle for half or less of the amount he’s asking for (depending on his character, sense of grievance, and how much he’s genuinely suffered and lost as a result of the accident). If some or most of that is covered by your insurance company, you won’t have come out of it too badly, in my opinion.
Finally, to answer the question in your original post, you may rest assured that, whether or not this goes to court, it will not have any bearing on your visa or work-permit status.