This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider potentials given by general radial nonnegative kernels which are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Using the integrability of those kernels, we rewrite the problem as the minimization of the difference between the classical perimeter and a nonlocal perimeter. Renormalizing to a fixed volume, and noticing that the parametrized family of functionals localizes to a multiple of the classical perimeter as the mass goes to infinity, we show that, if the first moment of the kernels is small enough, the problem admits minimizers of arbitrarily large mass. In addition, we prove that, after renormalization, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of the ball, and show that our upper bound on the first moment of the kernels is a sharp stability threshold for large balls. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.