Hello!

When I try to ping another host on the LAN it usually works the first
time, and then stops working because the other host does not replies to
ND requests. 

It would be better if you gave full tcpdump where "working" stage
is present too.

Apparently verything works again if I switch the interface
in promiscuous mode.

At receiver? If yes, then it is most likely wrongly configured
multicast filter at device. Though it is still not clear why it worked
the first time.

Alexey