It does look like the channel pulses are not sequential so you will probably need to use the multiplexer with your receiver. The challenge is to get the switching of the multiplexer synchronized so that you don't miss part of a pulse. To do this you need to ensure that you only switch the multiplexer to the next channel after that channels pulse goes low (that's the easy part) and you need to wait before taking the next pulse width measurement until you are sure that channels pulse has not already gone high.
One way to do this is to take two readings for each channel before moving on to the next one. It won't matter if the first measurment was in error because it will be overwritten by the second. The disadvantage of this technique is that it takes twice as long for changing pulse widths to be recognized but that may not matter in your application.
If you use this method then you can ignore the existing code to detect the sync pulse and replace that with a counter that increments the channel counter after the second trailing edge is detected.
If that isn't clear then I will try to post some pseudo code.
Or, if it is ok to miss some pulses then perhaps you should reconsider using pulsein with the multiplexer. The logic would be similar to the above: Wait for the trailing edge of the previous pulse, then store the next pulse in the channel array, increment the channel and repeat.
The advantage of the interrupt version is that it will never miss a pulse edge no matter what your code is doing but it will only work if the pulses are sequential. As yours are not than perhaps the simpler PulseIn route is better for your app.