We know that in order to describe spin 2 massless particles quantum mechanically we need to go beyond the field theory and consider the consistent quantum gravity theory (i.e. superstrings), but to be able to describe the massive spin 2 mesons, I do not see the need to go beyond the field theory. Is this because they are composite? Does it mean that we can not describe elementary spin-2 massive fields in field theory? Well, I know that it has been showed that gravitons can not be composite. Is this the reason?
In QFT any massive particle with a spin $\ge$ 1 will be non-renormalisable. Even a W boson would be non-renormalisable except of course that it acquires mass through the Higgs mechanism and is massless above the weak transition energy.
And this is the point. A massive particle of spin $\ge$ 1 (or a massless particle of spin > 1) will be non-renormalisable unless some other physics intervenes before the interactions become infinite. In the case of the W boson the Higgs mechanism intervenes. For a massive spin 2 composite particle there will be some energy at which the particle breaks up. For an elementary spin 2 particle (massive or otherwise) a QFT description will never be renormalisable.