If I understand right you basically want the color bar on the right under the dates to match the color of the client name. you will need to create a new rule for each client with the conditions for creating the bar and with $A1=client1 and set the format color for client1 (note you can include in the formula a condition to always include the client column A in that rule and hence let conditional formatting color those cells too). So something like:
Applied to A1:ZZ10000
Custom Formula: =("Client1"=$A1)*((Column(A1)=1)+( formula for date columns ))
