S-boxes with non-linearity greater than 112 are considered prone to side channel attacks. If a non-linear S- box with non- linearity greater than 112 (excluding bent functions with non-linearity of 120) is implemented through lookup tables, then the amount of power consumption should be similar to 112 S-box. In other words, side channel attacks shouldn't be able to distinguish between the two S-boxes. The question is, if an S-box with non-linearity greater than 112, implemented through look ups is safe or not?
I think you have mis-understanding and misconception of how nonlinearity of sbox influences on side channel.
The 112 is nonlinear value of good properties sbox (such as AES) , I have not seen sbox with greater value in my literature.
I will go beyond you question, to evaluate s-box against side channel attack such as differential power analysis (DPA), other parameters are taken : Transparency order (TO) (the lower the TO, provides higher resistance to DPA) and signal to noise ratio (SNR) ( the lower the SNR ,provides higher resistance to DPA).
Prouff showed in his paper (DPA Attacks and S-Boxes), the highly nonlinear s-box are prone to DPA and the construction of highly-nonlinear S-boxes with small transparency order (close to 0) is an open problem.
I recommend you download the SET tool from this link and start to play it with different constructions of Sbox to gain more knowledge