Proof for collision resistance on a block-based hash
If I understand you correctly, you are using the Simon key expansion process, so that the input to your block is the 256 bit Simon key, and the output is the 256 bit last round key (and you're ignoring the Simon encryption process entirely).

If so, three nits:


I believe that you can find preimages for your hash function in $O(2^{64})$ compression function evaluations.  This happens because the key expansion process is invertible; that is, given a 256 bit output, you can find the corresponding 256 bit input.  So, what an attacker would do is assume a 2 block message; he would:


Generate $2^{64}$ initial blocks, and compute the intermediate states from them.
Generate $2^{64}$ final states consisting of the target hash and $2^{64}$ different $K_i, K_{i+1}$ words, and compute the inverse key expansion
Look for a collision in the $K_{i+2}, K_{i+3}$ words; if you find one, this implies a preimage


This is easy to fix; make the transform noninvertible.
If you want your hash function to be collision resistant, you'll need to add some collision-free padding mechanism. 
This architecture is actually closer to a Sponge hash function than a Davies-Meyer type construction (just in case you're wondering)