There are two misconceptions in the statement of your question.

1. When you say "If the we tune these parameters to special values, namely the fixed point, we can observe the scale invariant behaviour" you are confusing critical point and fixed point which usually are two different points in theory space where the RG acts, see my previous answer: Critical 2d Ising Model
2. One does not perturb a fixed point by acting on it with the RG (which does nothing by definition) but by manually moving slightly away from it (en then acting with the RG), i.e., by adding a new term in the action. For example the $\phi^4$ model is obtained by adding a quartic term to the Gaussian fixed point.

There are two misconceptions in the statement of your question.

1. When you say "If the we tune these parameters to special values, namely the fixed point, we can observe the scale invariant behaviour" you are confusing critical point and fixed point which usually are two different points in theory space where the RG acts, see my previous answer: Critical 2d Ising Model
2. One does not perturb a fixed point by acting on it with the RG (which does nothing by definition) but by manually moving slightly away from it (and then acting with the RG), i.e., by adding a new term in the action. For example the $\phi^4$ model is obtained by adding a quartic term to the Gaussian fixed point.