  Inelastic neutron scattering has been used to study the in-plane Cu-O
bond-stretching mode in oxygen doped La$_{1.94}$Sr$_{0.06}$CuO$_{4.035}$ ($T_c
= 38\,\text{K}$) and La$_2$CuO$_{4+\delta}$ ($T_c = 43\,\text{K}$). Similar to
results from optimally doped La$_{1.85}$Sr$_{0.15}$CuO$_4$ ($T_c =
35\,\text{K}$), we observe anomalous features in the dispersion of this
half-breathing mode in the form of a softening halfway through the Brillouin
Zone. Considering the differences in electronic structure and local environment
between the oxygen- and strontium-doped compounds with similar $T_\text{c}$, we
rule out a connection between the phonon anomaly and structural instabilities
related to the specific dopant type. We interpret the phonon anomaly as a
signature of correlated charge fluctuations possibly connected to stripes.
