First and foremost a Beta 58 is a dynamic supercardioid microphone it does not need or tolerate any external power. I will admit it is one of the best vocal mic you can purchase. Any DC applied to it will cause distortion and possibly burn it out. Your Beta58A drawing shows that it is transformer coupled. You will need a Differential front end to amplify the microphones output, not single ended as in your drawing. You will need about 24 bit analog converter to pick up the quality of the microphone and convert it to digital where you will have to do processing. Multiply the A/D bits times the conversion rate required times 3. For 44Khz 24 Bits 3 channels you need to process 3,168,000 bits per second or 395000 bytes a second. A sigma delta converter may work better for you. Try this link: https://www.ti.com/europe/downloads/Choose%20the%20right%20data%20converter%20for%20your%20application.pdf
Take a look at this link: https://www.rane.com/note137.html it will give you some background information. The simplest solution I can see at this point is to get a mixer that you can gate the outputs. These will have the ability to direct each channel to an effects processor (FX). Connect that to a AC/DC conversion circuit via a 600 Ohm isolation transformer. You can then read the output of the conversion circuit with the arduino A/D inputs. Then do some software to determine which is to be turned on and control the gates accordingly. You can use a inexpensive condenser microphones placed close to the Shure and use them for level measurement only,
Good Luck & Have Fun!
Gil