Something like this is needed. Just to mention one reason: the pointwise sampling variance of the density estimate typically depends on both the bandwidth and the shape of the kernel (see Silverman 1986 ch. 3.3)

I was thinking of instead changing the UnivariateKDE object to include the kernel, something like:

mutable struct UnivariateKDE{R<:AbstractRange} <: AbstractKDE
    "Gridpoints for evaluating the density."
    x::R
    "Kernel density at corresponding gridpoints `x`."
    density::Vector{Float64}
    "Kernel"
    dist::UnivariateDistribution
end

Then, the output of the kde_lscv() could be
UnivariateKDE(k.x, dens, dist)