Interpolation space between $L^1\cap L^2$ and $L^1$

In the case of a set of finite measure (Bourgain in the quoted paper deals with the case of finite measure (torus)) we have that $\Vert f\Vert_1\leq C\Vert f\Vert_2$ so we actually have $(\infty,1)$ and $(2,2)$ estimate and therefore we have $\Vert Tf\Vert_{p'}\leq C\Vert f\Vert_p$, $1<p<2$, by Marcinkiewicz or Riesz-Thorin.

Edit: This is an answer to an earlier version of the question which was not so detailed and only after a discussion with the author of the question details regarding the dependence of the constant were added so you should take it into account before you decide to downvote my answer.

As requested, I post my comment as an answer (although this is not a true answer, just a possibly useful reference; feel free to edit it if this approach works out).

In Section 3 of the article Interpolation of sum and intersection spaces of $L^q$-type and applications to the Stokes problem in general unbounded domains, P.F. Riechwald studies interpolation between spaces $L^2 \cap L^q$ for $q \geqslant 2$ (and also $L^2 + L^q$ for $q \leqslant 2$). His main tool is Theorem 4, which I reproduce below.

Theorem. Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $f \in L^1(\Omega) + L^\infty(\Omega)$ be a given and fixed function. Then there exist linear maps $$ S_1 : L^1(\Omega) + L^\infty(\Omega) \to L^1((0, 1)) , \qquad S_2 : L^1(\Omega) + L^\infty(\Omega) \to \ell^\infty $$ and $$ T_1 : L^1((0, 1)) \to L^1(\Omega) + L^\infty(\Omega) , \qquad T_2 : \ell^\infty \to L^1(\Omega) + L^\infty(\Omega) $$ satisfying the equality $$ f = T_1 S_1 f + T_2 S_2 f $$ almost everywhere. Moreover, these maps satisfy the estimates $$ \|S_1 u\|_{L^p((0, 1))} \leqslant \|u\|_{L^p(\Omega)} , \qquad \|S_2 u\|_{\ell^p} \leqslant \|u\|_{L^p(\Omega)} $$ and $$ \|T_1 u\|_{L^p(\Omega)} \leqslant \|u\|_{L^p((0, 1))} , \qquad \|T_2 u\|_{L^p(\Omega)} \leqslant \|u\|_{\ell^p} $$ for all $1 \leqslant p \leqslant \infty$ and all $u$ in the respective $L^p$-spaces.

I suppose the argument used in the proof of Theorem 3 implies that the complex interpolation space between two spaces $L^1(\Omega) \cap L^p(\Omega)$ is again a space of this form. This, in turn, should imply the desired bound on $\|T f\|_p$.