Is such a crypto-system available? $E(k_1, pt) = c_1, E(k_2, c_1) = c_2, D(k_{new}, c_2) = pt$, where $k_{new} = f(k_1, k_2).$ Sharing $k_{new}$ and $k_2$ should reveal no information about $k_1$.

Clarifications:


Being able to decrypt the ciphertext $c_2$ knowing only $k_{new}$ is the first desired property of the system
One cannot derive $k_1$ knowing both $k_{new}$ and $k_2$.
One cannot easily derive $k_1$ knowing other pairs $k_2'$ and $k_{new}'$. See @poncho's comments.


$D(k_{new}, c_2) = pt$ or $E(k_{new}, c_2) = pt$ doesn't matter as long as you recover the plaintext.

Real world example to clarify things even more:


Alice has a very large database which it chooses to store at Eve's site. Because Alice doesn't want Eve to read the data, nor doesn't she know in advance which data will be shared with whom, she encrypts all the records with a single key ($k_1$).
Now Bob requests access to some specific information. Bob and Alice know each other so Alice gives him access to that subset. However Alice wants to do that efficiently. Also Alice doesn't want Bob to be able to read anything else except the shared subset. Therefore she cannot: a) Decrypt the shared subset at Eve's site and then re-encrypt with a different key (this would expose all the database contents to Eve), b) Retrieve the data from Eve, decrypt, re-encrypt and store back data at Eve's site - this implies a round-trip delay which is prohibitively expensive.


Thus Alice needs a cryptosystem which would allow re-encrypting the data directly at Eve's site, however Eve shouldn't be able to read any plain text (nor the one shared with Bob, nor the rest of the database records). Sharing the decryption key ($k_{new}$) is done through a direct channel between Alice and Bob (e.g. using Diffie-Hellman).

The "cryptosystem" stated in the question is just a way I saw that happening. Note that $k_1$ shouldn't become known to Bob (e.g. if Bob colludes with Eve or other people with whom Alice shares (possibly different sets of) data).
This can be done with RSA where k1 and k2 are encryption keys and f is  multiplication modulo the totient of the public modulus.
The last property, that knowledge of other pairs $k_2'$ and $k_{new}'$ does not reveal $k_1$ is satisfied by using a different modulus for each pair generated.