How can I estimate the amount of entropy (with JavaScript) that I'm seeding a CSPRNG with?
If someone chooses to run your program with a seed that you can't predict but an adversary can, you won't be able to tell.  There is no way around this: nothing about the software can tell you what an adversary does or doesn't know; you can only use cryptography to ensure that if an adversary doesn't know a key and the message, then they can't decrypt the message.

In general, to estimate the entropy of a system, you need to have a model for how the system behaves.  This is how all entropy estimation tools work: they posit some parametric model for how the input data might have been generated; then use the data to fit parameters for that model; and finally spit out the entropy of the model with the fitted parameters.