Don't know how secure deterministic asymmetric encryption (lets call it DAE) is, but you want it to fill out exactly the role for which hashes aka trap-door functions were invented.
I'm actually working in a facility for cryptographic research and I asked someone who should know the actual research in the field. He told me that there is not much research about the strength of DAE, I suspect because there are much better alternatives.
Usually you would do the following: Store the data with symmetric encription and a random key that is encrypted with asymmetric encryption (this method is standard for any data bigger than a few hundred bytes, since asymmetric encryption is very slow). Beside this store the hash of the data. At the moment sha256 seems to be the hash everyone uses, with sha512 for the really paranoid.
As you can see this fulfills all your requirements