Note that particles are not just for perturbation theory. There is a particle whenever there exists a particle annihilation/creation field configuration. A proton is a particle so writing down its creation/annihilation field configuration is in theory possible, though maybe not in practice.
Another point is that infinities do not necessarily make mathematicians cry. Abraham Robinson is quite pleased with them. It seems a possible hypothesis that at least some QFT are mathematically well-defined using non-standard analysis. Where 'some QFT' at least renormalizable and perhaps also asymptotically free. I don't know enough about it to know how the Haag theorem, mentioned in another comment impacts this.
Another analogy (flawed as any of them). Sports teams "exist" in a sense. They meet one another in well-defined interactions, called matches, and such an interaction can be described as if teams were well-defined atomic entities, producing a score.
But a sports team is not atomic, not a "final reality" entity. A sports team can pass through one gate, or through several gates, when entering a stadium. From a doctor's perspective, the team "does not exist", a doctor only operates in terms of individual players' organisms.
Another point is that infinities do not necessarily make mathematicians cry. Abraham Robinson is quite pleased with them. It seems a possible hypothesis that at least some QFT are mathematically well-defined using non-standard analysis. Where 'some QFT' at least renormalizable and perhaps also asymptotically free. I don't know enough about it to know how the Haag theorem, mentioned in another comment impacts this.