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I am looking for an asymmetric cipher which has the property that very short inputs result in very short outputs.
.NET’s standard RSACryptoService seems to output a minimum of 128 bytes for any short plaintext, including the empty plaintext. Minimum supported BlockSize is 128 and the only supported Mode is CBC.
The perfect algorithm would not expand the plaintext at all, while providing the asymmetric property. Expected input size is 50-100 bytes, certainly never more than 150.
It is certainly possible to achieve your goals for the sizes you are interested in by using elliptic curve ElGamal encryption and variants. The Bouncycastle C# library is one option for implementation.
So for example by using the NIST P-256 curve you can encrypt 32 bytes of data and get a 64 byte result. If you use proper padding you can encrypt less data. I believe PGP once considered implementing this but I don’t know if they ever did.