In theory a T-test is only valid where the statistic you are testing is expected to be be normally distributed. Its not clear to what extent this is true for log2Fold changes, particularly if we don't know how they were produced. I suspect that these LFC for highly expressed proteins probably are fairly normal, but that there is a departure from normality when counts are low.

However, in practice, the t-test is fairly robust to departures from normality, and is usually more powerful than the non-parametric alternative, even when the normality assumptions are violated. We don't usually use unmoderated t-statistics on gene expression data, because estimating the variance can be difficult with the low sample numbers usually available, but with 200 samples you should be fine. Although with 200 samples, you are probably also fine, powerwise, with the Wilcox.

One thing I would watch out for however, is your assumption that the expected LFC under the null hypothesis is 0. This will only be true if the data is well normalised, and its not clear from your description what normalisation may or may not have been applied to the data before the calculation of LFCs. On the assumption that most genes are not DE, you could look at the distribution of mean LFCs across the whole dataset and check that it is symmetrical and centered around 0.