Possible issues with ECIES using a static (non-ephemeral) sender key
What trade-offs exists with this alternate approach?


Unless you need the non-repudiability you produce a larger ciphertext in a slower way for no interesting gain in security when signing and encrypting over using the static keys (apart from the "other downfalls" below).


  Are there any other downfalls of this approach?


Indeed if now either of your long term secret is compromised all ciphertexts can be decrypted unlike in traditional PKE schemes where only one party has such a long-term secret.


  Is this regarded as just a form of ECIES or is it possibly something different since the sending key is not ephemeral?


Technically speaking it doesn't fit the traditional signature of a public key encryption scheme because there the encryption isn't allowed to maintain state and obviously a sender secret can't be part of the public key. The most common name for this I know would probably be crypto_box (taken from NaCl's naming) which does essentially this. Academically speaking the correct name is "public-key authenticated encryption" because it is the public-key equivalent of authenticated encryption.