I'll stand on my comments that your clip lead and external power
supply kludge is probably causing problems. Find some suitable
electrolytics and place them across where the battery would normally
connect. The idea is to reduce the impedance that the counter sees in
the direction of the power supplies. That should reduce the
motorboating (oscillation).

I tried putting a 100uF cap in that area, the result was that it greatly
increased the frequency of the stray clicks....
I wonder if a row of 10 9-volt batteries would work?