Is GAP NP-hard with at most two balls per bins?
You can reduce numerical 3-dimensional matching (N3DM) to your problem.

Given an instance of N3DM $X\times Y\times Z$ with the bound $b$, say $X=\{x_1,\ldots,x_k\},Y=\{y_1,\ldots,y_k\},Z=\{z_1,\ldots,z_k\}$, you can construct $k$ bins with capacities $b-x_1+M+M^2,\ldots,b-x_k+M+M^2$, and $2k$ balls with sizes $y_1+M,\ldots,y_k+M, z_1+M^2,\ldots,z_k+M^2$, where $M$ is a very large number. Now there is a valid solution to the N3DM instance if and only if you can pack all the $2k$ balls.