Software debounce for WC8 counter?

Increase the horizontal scale to 20 milliseconds per division and the horizontal scale to a lot less to get more resolution on the pulse height.
What are you triggering the 555 with?  If you are using a pushbutton switch you should press very quickly and see a nice square wave (the waveform should return to zero once you have the correct horizontal time scale).
Not sure on the output.  You would just disconnect the output pin from the circuit and measure that directly with the scope.
Is the scope set at 50 ohm input termination?
Just want to provide an update.
I tested the breadboard prototype and after fixing a circuit assembly mistake the circuit worked perfectly.  A nice square wave on the output, and no bouncing counts.
I left the LED in the circuit because the output voltage (3.2v) from pin 3 of 555 seems to be enough for the WC8 to reach threshold for HIGH on the TTL input.  And the LED serves as a great status indicator.
Turns out the reed switch is closed for almost a second with each 'pulse'.  The pulse counter completes one cycle every 2 to 3 seconds.  So 200ms is fine in terms of debouncing.
Once the breadboard prototype worked I then made up a soldered version (photo attached).
Thanks again to Larry, Robbin and CAI for you input.  I hope other people find this thread useful if they experience bouncing issues with the TTL counter on the WC8.