We don't know that helmet use in and of itself has a direct bearing. It is likely that risk compensation influences these figures, but we do not have any idea how much.
Apart from the research finding they do increase the accident rate, this may be due to several factors, as you point out. We may never be 100% sure how much each factor contributes. The best evidence probably comes from NZ where both distance and time cycling estimates have been published for before legislation to after.
Robinson's submission to the Au Senate included;
H) Increased risk of injury per cyclist since helmet laws were introduced
Several analyses have compared numbers of injuries with the numbers of cyclists. They all suggest that injuries per cyclist have increased from what would have been expected without helmet laws.
In New Zealand, from 1989 to 2011, average time spent cycling (on roads and footpaths) fell by 79% for children aged 5-12 (from 28 to 6 minutes per person per week) and 81% for 13-17 year olds (52 to 10 mins/person/week).
Adult cycling declined from 8 to 5 minutes/person/week then trended back up to 8 minutes. Graphs of cycle use over time provide strong evidence that the requirement to wear a helmet discouraged cycling. The reductions in cycling were accompanied by increased injury rates. Between 1989 and 2012, fatal or serious injuries per million hours of cycling increased by 86% for children (from 49 to 91), 181% for teenagers (from 18 to 51) and 64% for adults (from 23 to 38).
All this type of information takes some explaining, so a simple reply that helmets increase the accident rate by 14% is short but reflects serious doubt about the safety merits of helmets.