You are correct about the delta method and its assumptions, that is why the delta method is considered an approximate method for computing precision.
If you are uncomfortable with the assumption of independence between components in distance sampling analysis, the preferred solution is to use a nonparametric bootstrap to measure uncertainty in your density estimates. Both the Distance for Windows software as well as the Distance R package contain routines for constructing measures of precision using bootstraps; usually by resampling transects (points or lines) with replacement.
Consult Section 5.7.2 of Buckland et al.
(2001) for a discussion of bootstrapping in the context of point
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