Using ChaCha20 as a PRNG with a variable-length seed
Davies–Meyer?  That would be a very unusual way of seeding a PRNG.  The standard way of entropy extraction is to use a vetted hash algorithm such as SHAx.  I'm not quite sure how you want to use your RNG, and I'm having trouble deciphering how many bits of what you want.  I'm going to assume that you simply want 256 bits of entropy to inject into the state.  

Having an entropy source available changes things somewhat compared to a plain PRNG. You don't need a key or a nonce in this case.  You'd collect 128 bytes from your entropy source.  It's twice 64 as NIST recommend that you double the input entropy for each bit of output entropy when using a hash based extractor.  Then run it through SHA256 and inject into ChaCha in place of the nonce and key.

Further I'm also assuming that this is to be only a PRNG.  With seeding from a true entropy source, you won't be able to repeat the output sequence.  Intuitively it feels wrong to use a PRNG to self seed, but I can't demonstrate this mathematically.  Follow the hash route as it's the obvious way, well proven and likely to be simpler than rolling your own recursive version of Davies-Mayer.