ElGamal for ECC - Not working with Edwards curves (ed25519)

I know the usual way of using encryption/decryption with ECC is DH, however, this only works with two keypairs of exactly the same kind, for example two curve25519- or two p256-keypairs.

With ElGamal, one can get a symmetric key for a recipients public key no matter which kind of key they are using themselves.

This works great if users don't have the same kinds of keys, and it doesn't use the same shared secret each time.

However, it only seems to work for Short type curves - NIST, SECG, brainpool and so on. (My javascript implementation(1) is based on SJCL's implementation, but uses "elliptic" as a library, which supports Short, Montgomery and Edwards curves.)

Is there any change to make it work with Edwards type curves like ed25519? Or is there any reason this actually isn't possible?

Reasearching on this topic doesn't bring much detailed information(, unless you get the exact math behind Edwards curves, which I don't at the moment, I'm just using the elliptic library for a simple app).

Thank you for your answers :)

(1) Some pseudocode (simplified):

Get a symmetric key for a known public key (to get a symmetric key only the recipient will be able to find out knowing a public key):

ec = curveobject(curve_the_pubkey_is_on);

secret = random_secret_key();

tag = ec.g.mul(secret); // pass along with the message

key = public_key.mul(secret); // use for symmetric encryption

Get a symmetric key for a known tag (to get the symmetric key knowing the tag

and own private key):

key = tag.mul(private_key); // can decrypt message with