Doing the Impossible
Not impossible, just difficult. Difficult we do right away. Impossible just takes a little longer.
You could use a Quad 2-Input Mux such as the 74LS157 to set the counter values for the divide by 6 counters. The Mux SELECT input is controlled by the J7 Preset/Load switch. This allows selection of either the load switch values or the divide by 6 values for the Counter Preset. (The Mux is not needed for the counters that go to 0 and rollover to 9.)
The Mux Select input goes to J7. The Mux I0 DATA inputs go to the Counter Preset Switches and the Mux I1 data inputs go to the desired count preset value for a divide by 6. The Mux outputs go to the counter DATA inputs. In the J7 load position (Mux Select Low), the counter presets are thus connected to the switches. In the J7 run position (Select High) the presets are set to the hardwired values to cause the proper divide ratio. For example the 10's minute digit would be loaded to binary 5 on the next count after it reaches zero.
Since the counter load control is asynchronous you will need a D flip-flop to delay the counter load signal until the count following 00 for the counters you want to divide by 6. Otherwise it would instantly load 5 when it reached 00. Connect the RCO signal from the counter to the D FF input and the FF Q output to the counter LOAD input. The FF clock should be 2Hz for proper sync of the LOAD signal.
You will also need to add 2-input AND gates at the divide by 6 counter LOAD inputs. The AND inputs are the FF output and J7 so either can control the LOAD function for the divide by 6 counters.
Hope you can follow all that. I could draw if for you but what's the fun in that. As they say in the textbooks, that's left as an exercise for the reader. It's not a trivial addition to the circuit but, I think, doable.