DES - encrypt 3 times - DES(DES(DES, k), k'), k) - stronger than 2DES with 2 different keys? What if were to use DES in the following way:

For a given plaintext x and key k we will perform: DES(DES(DES, k), k'), k), where k' is the complementary of k. 

If an adversary has a small amount of text and their ciphertexts, would he still have to run 3 * 2^56 encryptions: for each key perform DES(DES(DES, k), k'), k) and therefore this encryption is stronger than 2DES with 2 keys (meet in the middle requires 2^57 runs) but less than real 3DES which is much more than 2^57 ?

Or is there a smarter way ?
As pointed out in the comments two key triple DES (without your complementary key condition) already has efficient attacks against it, going back to the 80's. See for example a detailed analysis in Van Oorschot and Wiener here which converts a chosen plaintext attack by Merkle and Hellman to a known plaintext attack.

Since 2 key 3DES is already weak, trying for sophisticated and complicated improvements based on the complementary key condition is likely not worth the effort.