Can bright LEDs cause eye damage even if they are dimmed with PWM?
It is permissible within certain limits. The best place to look is probably the associated IEC standards (IEC 60285 Laser Safety and IEC 62471 Lamp Safety), which are generally internationally recognized as best practice. Unfortunately I can't post excerpts of them here since they're copyrighted.

Choosing which standard to apply depends on how the LED is used. OSRAM has a very comprehensive appnote describing how these standards apply to infrared LEDs and how to calculate permissible exposure.

Your particular case focuses on pulsed light. In general, PWM'd light is weighted against its averaged value, so long as the individual pulses do not exceed an irradiance limit (given by a chart in the standard of pulse length vs. irradiance). This is all outlined in the OSRAM appnote, although since you're in the visible range you'll have to refer back to the source standards to see what the particular limits are for your wavelengths.

Edit: Found another appnote which may be useful to you - OSRAM has an appnote on 62471 as a whole, not just IR.

The best place of course to look is the standard itself, but it costs about $250. If this is a product you're designing, that's probably worth it, but if this is just a hobby project I would scavenge information based on appnotes.