CrossRoads:
Easily.
You can do it with SPI.transfer() commands to set up the MAX7219 and send it the 6 digits you want to display, or use some library to do it.
This is a much larger version that uses a Promini for control and a MAX7221 (MAX7219 is identical but with faster signal edges) driving countdown timer display (and countup is easier) in the middle and 2 set of 00-99 digits at the upper sides.
The '328P based promini does the timing, no external clock needed.
I think your hardest part will be making it all watertight.
Can make it rechargeable-battery powered with sealed connectors for the pushbutton for the lap counter and the external reset.
Or go fancier and make the reset wireless, so the only connection is for the lap button.
CrossRoads, you really do understand what I am trying to design! I thought I was explaining it wrongly. In the image you attached, I see the two couting LEDS on the left and right as well as the bottom set couting real time. What at the 2 digit 7 segment LEDS under the large left do?
Is this image something you have made or something you found online?