  We report a comprehensive Cu $K$-edge RIXS investigation of $\rm
La_{2-x}Sr_xCuO_4$ (LSCO) for 0$\leq$x$\leq$0.35, stripe-ordered $\rm
La_{1.875}Ba_{0.125}CuO_4$ (LBCO), and $\rm La_{2}Cu_{0.96}Ni_{0.04}O_4$ (LCNO)
crystals. The RIXS spectra measured at three high-symmetry momentum transfer
(\textbf{q}) positions are compared as a function of doping and for the
different dopants. The spectra in the energy range 1-6 eV can be described with
three broad peaks, which evolve systematically with increased doping. The most
systematic trend was observed for \textbf{q}=($\pi$, 0) corresponding to the
zone boundary. As hole doping increased, the spectral weight transfer from high
energies to low energies is nearly linear with \emph{x} at this \textbf{q}. We
interpret the peaks as interband transitions in the context of existing band
models for this system, assigning them to Zhang-Rice band$\rightarrow$upper
Hubbard band, lower-lying band$\rightarrow$upper Hubbard band, and lower-lying
band$\rightarrow$Zhang-Rice band transitions. The spectrum of stripe-ordered
LBCO was also measured, and found to be identical to the correspondingly doped
LSCO, except for a relative enhancement of the near-infrared peak intensity
around 1.5-1.7 eV. The temperature dependence of this near-infrared peak in
LBCO was more pronounced than for other parts of the spectrum, continuously
decreasing in intensity as the temperature was raised from 25 K to 300 K.
Finally, we find that 4\% Ni substitution in the Cu site has a similar effect
on the spectra as does Sr substitution in the La site.
