How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian
The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in QFT textbooks) and produce, e.g., by a point-splitting procedure, correlations with insertion of composite fields like $\phi^3$. You should then identify your theory via the equation of motion, e.g., $-\Delta\phi+m^2\phi=-\lambda\phi^3$ holding inside correlations. To see how this is done rigorously, see the article by Feldman and Rączka in Ann. Phys. 1977 or the recent article by Gubinelli and Hofmanová.
Also, an interesting example is the following. Let ${\rm Conf}_n(\mathbb{R}^2)$ denote the configuration space of $n$ points in $\mathbb{R}^2$, i.e., the set of tuples $(x_1,\ldots,x_n)$ made of $n$ distinct points in $\mathbb{R}^2$. Consider the functions $S_n:{\rm Conf}_n(\mathbb{R}^2)\rightarrow\mathbb{R}$ given by $$ S_n(x_1,\ldots,x_n)=\sqrt{\sum_q\ \prod_{1\le i<j\le n}|x_i-x_j|^{\frac{q_iq_j}{2}}} $$ where the sum is over "neutral configurations of charges" $q=(q_1,\ldots,q_n)\in\{-1,1\}^n$ such that $\sum_i q_i=0$.
One can show that $\forall n,\exists K_n>0,\forall (x_1,\ldots,x_n)\in{\rm Conf}_n(\mathbb{R}^2)$, $$ S_n(x_1,\ldots,x_n)\le K_n\prod_{i=1}^{n}\left(\min_{j\neq i}|x_j-x_i|\right)^{-\frac{1}{8}}\ . $$ The quickest proof I know for this inequality is following the optimal matching argument in Appendix A of "Complex Gaussian multiplicative chaos" by Lacoin, Rhodes and Vargas. Then it is not hard to show that for every Schwartz function $f\in\mathscr{S}(\mathbb{R}^{2n})$, the integral $$ \int_{{\rm Conf}_n(\mathbb{R}^2)}S_n(x_1,\ldots,x_n)\ f(x_1,\ldots,x_n)\ d^2x_1\cdots d^2x_n $$ converges and defines a temperate distribution in $\mathscr{S}'(\mathbb{R}^{2n})$. This is explained in Section 2 of my CMP article "A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion".
Now it is a fact that the resulting distributions $S_n\in \mathscr{S}'(\mathbb{R}^{2n})$ satisfy the Osterwalder-Schrader Axioms, and therefore can be analytically continued into Wightman distributions satisfying the Wightman Axioms and thus via a GNS type construction, give in the end a quadruple $(\mathcal{H},U,\Omega,\phi)$ obeying the Gårding-Wightman Axioms. The $S_n$ are also the moments of a probability measure on $\mathscr{S}'(\mathbb{R}^{2})$.
Quiz: What is the Lagrangian of this theory?
I will come back later with an answer, but regarding the construction via probability measures, I explained this already so I will not repeat myself and refer to
Reformulation - Construction of thermodynamic limit for GFF
A set of questions on continuous Gaussian Free Fields (GFF)
A roadmap to Hairer's theory for taming infinities
Quiz answer: It is the Ising CFT. Note that I tried to see if one has an equation of motion of $\phi^4$ type but my computations got out of hand rather quickly when looking for an explicit $\phi^3$.