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.).

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. This tells you that this theory is not valid upto every high energy scale.

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, you must find that it's need arise as a redefinition of some quantities which in this case will not be finite in number ( like field strength, mass etc.).