The interplay of intralead and dot-lead Coulomb interactions on shot noise of a quantum dot coupled to Luttinger-liquid leads is systematically investigated in the Kondo regime. By equation of motion and canonical transformation methods, a general shot noise formula with transmission coefficient is derived, the factors of the above two types of interactions and on-dot Coulomb interaction being taken into account. Then we study in detail the shot noise and Fano factor as a function of bias voltage. Numerical results show for weak and moderately strong intralead Coulomb interactions, the dot-lead interaction not only helps to improve the shot noise, but also narrows the zero-bias Kondo dip. A series of polaron peaks appear in the differential shot noise when the renormalized parameter Y<1, a combination effect of the intralead and dot-lead Coulomb interactions. For Y>1, a novel type of excitonic dips appear only in the case of moderately strong intralead interaction. In the very strong intralead interaction limit, all features of excitonic peaks or dips disappear, and shot noise shows power-law dependence on the voltage with the renormalized exponent Y. The Fano factor is always sub-Poisson noise no matter whether Y is greater or less than 1.