Let $X_0,X_1,Y_0,Y_1$ be Banach spaces and $T \colon (X_0+X_1) \times (Y_0+Y_1) \to \mathbb{R}$ such that $$| T(f,g) | \le \| f\|_{X_0} \|g\|_{Y_0}$$ and $$| T(f,g) | \le \| f\|_{X_1} \|g\|_{Y_1} + C_1$$ for some constant $C_2>0$. We may assume that $T$ is bilinear.
I am interested in whether one can interpolate these two inequalities in the sense that $$| T(f,g) | \le \| f\|_{X_\theta} \|g\|_{Y_\theta} + C_2$$ for some constant $C_2>0$, where $X_\theta,Y_\theta$ are suitable interpolation spaces between $X_0,X_1$ and $Y_0,Y_1$ respectively.
By a simple scaling argument, it is clear that we can choose $C_1 = 0$ in the second inequality. Indeed, since $T$ is bilinear, we have $|T(f,g)| = 1/n |T(nf, g)| \le \|f\| \|g\| + C_1 / n$. Now you can pass to the limit $n \to \infty$.
The rest is clear.