In my current configuration and when changing the performance, I always need to activate / deactivate the "KbdCtrl" Switch on the 1st part of the relevant performance... but I need this to work automatically... Can anyone help here please? Thank you so much!!

Simply set it up with your KBD CTRL = Off On Part 1, then [STORE] your custom Performance.
The condition of the KBD CTRL setting is a storable parameter.

Your Performance will be directed to the USER Bank.
Suggestion: Take the opportunity to rename your version of each Performance.

Another approach could be to set your LMK3+ to transmit on channel 9, and put the MODX piano you want it to trigger in Part 9. (Any of 9-16 would work.) Advantage is that Kbd Ctrl will be off for this sound automatically. Disadvantage is that you lose seamless sound transitions between performances (even if they don't exceed the 4 part limit for SSS). At least that's how I understand it.

Is that alsoo possible using more layers/splits on your controllerkeyboard? Using part 9-16?

Yes, if your controller has the ability to transmit on multiple channels (zones), you can split/layer as many of MODX Parts 9-16 as you have zones, and they will play independently of whatever combination of Parts 1 through 8 that you are playing from the MODX' own keys.

If your keyboard lacks multiple MIDI outputs or otherwise is not able to give you the flexibility wanted - a computer, tablet, or dedicated hardware device (MIDI solutions or others) can be placed between each keyboard to handle duplicating a single MIDI channel to multiple MIDI channels.

For me, my goals were: portability, no (or minimum) extra gear, and no more expense than needed. MIDI solutions and like small hardware was more expensive and no more flexible than using a tablet (apple type) which I already carried to the gig for my setlist/tune reminders. I opted to purchase the Camelot solution which offered this routing capability as well as other features. A surface would have worked too as the same thing - a "tablet" with software that handled the routing (Cubase) which I already have. I opted for the iPad since I already had my sheet music loaded there. Also I feel the apple dedicated device is less likely to go belly-up than a PC. Unfounded - but part of the decision mix.

Another app that should do the trick is MIDI Breakout Box. Just 99 cents, and it will even work on an iPhone (you don't have to have an iPad).

But in my MacBook Pro, the applications are not working. I thought it may be the system fault so I consulted with apple technical service but they told me there is not any fault in the system. Kindly suggest me the way.

The Apple software suggested here thus far is focused on iOS - due to my bias of portability. The Macbook uses MacOS - and I'm not sure which applications have MacOS versions or not. I do not own a MacOS device.

I do know that Camelot Pro advertises a MacOS version. I find to re-channel MIDI is a little confusing using the app - but it does allow many ways and levels to do just this.

MIDI Pipe is something targeted for MacOS: http://www.subtlesoft.square7.net/MidiPipe.html

It has a channel routing feature among others.