3 momentary buttons controlling 3 relays
You have two very good reasons that you should use a microcontroller board to implement the logic for this.


You have momentary buttons and it takes a latching function to ensure that the corresponding relay for each switch stays on.
The logic to handle the exclusion between your three switches can be very easy to implement in the software of the microcontroller.
The microcontroller easily can adjust the switching dead time between relays.


The switches can very easily be implemented as low voltage directly into the microcontroller where software can then take care of debouncing the switches and implement a simple state machine that accepts the most recent switch press and only drives the singular relay coil that corresponds to that switch.

You will almost certainly require a driver circuit to buffer the microcontroller outputs that go to the relay coils. If you choose to use some off the shelf ready-made microcontroller board (arduino comes to mind) you can also find a ready made relay board that will work with the microcontroller and include the necessary buffer circuit right on the relay board.

The ready-made hardware is a great way to build up a one-off type project or to get a prototype concept proven out. If you are then actually making a production product you can then transcribe the prototype design into circuit boards of your own making to get to the desired product configuration.