14C bomb pulse dating seems to be since then a rather popular method in forensics as well judging by a quick google search.
Even with very low solubility there can be some dissolved transport into earlier or later sediments. For example, Woods Hole Oceanographic Institute has determined that plutonium mobility may be affected by oxidation state and that more mobile 244Cm can be transported before it decays to 240Pu.
Thus, anthropogenic radionuclides, particularly those of long half life and low solubility that were produced during atmospheric testing, have properties that make for pretty good stratigraphic markers. But the limitations should be kept in mind.
An additional problem is in defining the 'start' of the anthropocene as fitting your marker rather than the other way around. Certainly, people have had had geologic and ecological impacts before 1945. There are indications that aboriginal burning regimes over 20,000 years caused nutrient export with effects that can be seen in current groundwater chemistry.
have a explicit figure on the matter as well (Figure 1) showing a similar peak during the 1900's for measurement of beta radioactivity measurements in ice cores. Human made nuclear device explosions are a clear marker in this case. I assume this is only one case among many, as other posters in this thread pointed out.