The advantage of a Cox PH model is that it is designed to handle censoring, so long as the censoring is non-informative.* For example, the presence of censoring should contain no information about the actual time to event except that the event occurred later than the censoring time. Beyond that requirement there is no "acceptable limit" on censoring. For example, in a disease with low recurrence or mortality there will necessarily be a large fraction of censored cases in analysis of disease-specific survival. The power of your analysis will essentially be set by the number of events; with 200 events you should be able to evaluate a reasonably complex survival model with a dozen or so predictors.
What might be useful to allay your fears and those of your reviewers or readers would be to display the distributions of censoring and event times in a way that they can be compared, say as superimposed separate density plots. That, for example, could rule out a situation where most censoring times are earlier than most event times, a situation that might raise questions about whether censoring was really non-informative.
*It's not clear what test you used to determine that "censoring occurred at random," but non-informative censoring is not necessarily the same as random, as noted on the linked page. For example, say that everyone entered the study at the same time and the study was terminated at a fixed date, with those still in the study censored at that end date. That might not be found to be "random" censoring but it could still well be non-informative.
