Why is the subhash key (H) of AES-GCM defined as such?
In GCM, GHASH doesn't use K directly in order to ensure that if GHASH's key is compromised, the master key K remains secret. 


Actually, there's not why GCM was designed the way it was (it may be an effect, but not the reason).  Instead, the reasons were:


To allow provable security (based on the assumption that the block cipher was a secure keyed random permutation).  If we used $k$ as both the block cipher key, and used as an input to compute the tag, we need to make additional assumptions on the block cipher.
Because of the size mismatch; $h$ is 128 bits, $k$ is (at least, for AES) is either 128, 192 or 256 bits.  Yes, one could select an arbitrary section of $k$ as your $h$, however, that seems a bit, well, arbitrary...


In any case, to answer the question you did ask:


  If the mapping is universally constant wouldn't it be the case that this could be pre-computed and available somewhere?


Not really, the effort of pre-computing such a table would be at least $O(2^{128})$ operations; we believe that is infeasible for anyone, even as a one-time effort.  And, generating a rainbow table doesn't decrease this effort any; it just makes the generated table smaller...