I think this would depend on whether you have a legitimate substantial reason for why you think the error variance of that item is actually at the given value. Zero error variance sounds especially worrisome - in this case you are saying that the item is a perfect errorless indicator for the latent! (in which case, why would we need any other indicators?)
My personal feeling is that CFA isn't very meaningful with a single latent and three indicators - even with fiddling around fixing error variances and so on to get an overidentified model, the degrees of freedom will be so low that the model will inevitably fit "well". In such a situation you are reproducing the terms in the original covariance matrix with a model that is only very slightly less complex than the original covariance matrix, so it's easy for the model to "predict" the covariance matrix with small residuals. Also note that statistical power for fit statistics reduces dramatically when model df is lower.
EDIT: Just to expand on this a little:
In this case the reason you don't get fit statistics is not that the model doesn't fit well, but rather that the model is just as complex as the covariance matrix it is intended to explain, and as such fit is perfect (but meaningless, and therefore not reported).However with only three indicators I do not have any df's left so no model fit.