How can the 'randomness' of a finite sample of binary keys from a finite key space be estimated?
There are a few things to notice here:


any tested sequence is finite by definition, we cannot run tests on infinite sequences;
the size of the output is interesting, but if it is sufficiently random then the concatenation of the 20 bit sequences should also be random;
we cannot say for certain that a sequence is truly random, as many algorithms can generate output that seems random, but isn't - like a part of Pi from a particular offset;
peculiarities/bias/inter-bit-dependencies are not acceptable for key generation.


Great, that out of the way, let's advance to the answers:


the die-harder tests and any other tests seem fine for your particular problem;
the fact that you generate 20 bits at a time is not a particular issue, as long as it is random;
you should understand the properties of the random number generator and test to see if it is supposed to be random given those properties - basically you cannot and should not test for randomness without description of the RNG itself;
if you are afraid that your results are biased then you should at least whiten the generated stream and / or use it as input to a Deterministic Random Bit Generator (DRBG or CSPRNG).


As for (4), it seems you already assume that your protocol takes care of possible bias ( although you also talk about using the randomness as key data). In that case I would try and prove that the bias does not pose a problem. Furthermore, I'd make sure that then RNG cannot be used for other parts of the application that do require a cryptographically secure random number generator.

Finally, note that die-harder and similar suites work best on (very) large data sets. A few GB is kind of the minimum. I hope your generator is fast enough for that.



Caveat: I'm not a super-expert on this area and cannot answer about local properties your stream exhibits. I do think that the idea that the generic test suites won't pick up on this is flawed though. I'm pretty sure that suites like die harder will pick them up - but I'm not sure about the clarity of the returned results.