How to avoid copy and paste when two functions are very similar?

You don't have to make your parameterized selectChannels public. It can be a private implementation detail of both selectAlmostOkChannels and selectNotOkChannels, your public functions.

selectChannels can even be implemented as a function template, so that the generated code is equivalent to the hand-written "copy pasted" version, without the maintenance burden of code duplication

template<typename SelectFunction>
void ChannelSelection::selectChannels(int currentInkId, SelectFunction selectFn)
{
    bool selected = true;
    foreach (auto report, m_reports) {
        if (report.scoreByInk.find(currentInkId) != report.scoreByInk.end()) {
            auto tmpStatus = Assessment::getStatusFromScore(report.scoreByInk.value(currentInkId));
            selected = selectFn(tmpStatus);
            /* fill in */
        }
    }
    m_currentSelection.insert(currentInkId, selected);
}

void ChannelSelection::selectAlmostOkChannels(int currentInkId)
{
    selectChannels(currentInkId, [] (auto tmpStatus) -> bool {
        return /* fill in */;
    });
}

void ChannelSelection::selectNotOkChannels(int currentInkId)
{
    selectChannels(currentInkId, [] (auto tmpStatus) -> bool {
        return /* fill in */;
    });
}

You might have been taught that templates need to be in headers, but that's actually not the full story! Template definitions need to be visible where instantiated. Since your template is only used in the private implementation of your member functions, then your template definition can be in the same file that's implementing both your member functions