That 3 volt drop is probably much more that is so quick you cannot see it and enough to drop out the radio.
If you are measuring at the 13.8 volt point on the tuner plug it would be a good indication that is the problem.
Clean the fuses and all of the connections.?
If you had 13.8 volts and a 3 volt drop you are left with 10.8 volts and that is too low, especially since you are probably not seeing the total instantaneous drop, which is higher. The radio will see it and shut off.
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