The interaction term which you are using is $\phi^2\partial_\mu\phi\partial^\mu\phi$ ( add a coupling constant in front of this term), this term has by power counting mass dimension 6 hence the coupling constant has mass dimension -2. This reflects that the interaction term is non-renormalizable.
The best you can do is to make an effective field theory of it. The situation is similar to Einstein-Hilbert lagrangian, which is also power counting non-renormalizable.
None of your analysis will simply hold in QFT for this type of interaction. you can not just add any counter term you want, it must arise as a redefinition of some quantities which in this case will not be finite in number ( like field strength, mass etc.).