This is a hugely complex subject for which a categorical answer requires detailed river basin modelling and process calibration. The short answer is that in the short term many rivers deposit sediment much more rapidly than the hypothetical 1m per century, because sediment transport is roughly proportional to the fourth power of the stream velocity. So enormously high deposition occurs in the few hours per year of peak discharge. Mean annual deposition, however, is a different issue, and inordinately difficult to measure. A typical mean rate of sedimentary deposition is roughly 2 to 3 mm per year +/- at least an order of magnitude. This is slightly less than the current mean rate of sea-level rise, but remembering that the rate of sea-level rise is accelerating, and is likely to continue accelerating well into the next century. Projected rates of sea level rise by 2100 are tricky to estimate (too many unknowns) but >10 mm/year seems likely, in which case probably no significant rivers will be able to compensate by high sediment deposition. There will be occasional anomalies, such as the rivers around Mount Pinatubo in the Philippines, which had jaw-dropping sediment loads for the few years after the volcano erupted, caused by huge quantities of un-cemented volcanic ash over entire catchments, coupled with typhoon rainfall.
In the case of really big rivers with perennially high sediment loads, one thinks of the Mekong and the Ganges-Meghna-Brahmaputra, both of which have huge monsoonal sediment fluxes. Both river systems have pro-graded their deltas surprisingly recently. Check out their delta areas on Google Earth. All of that sediment was deposited during the last 6000 years. During most of that time, sea level rise seems to have been very modest, averaging about 1.5 to 2.5 mm/year.
Since you pose a climate-change related rate of sea level rise, one must also consider how climate change will affect the hydrologic response of these catchments, and hence how the rate of sedimentation will change? Even with a three-year research grant I would be hard-pressed to answer that question, but my suspicion is that sea-level rise will win, and most of the major deltas will succumb to the waves.