LCD Hotplug

Is it possible for an Arduino to be coded to respond to an LCD being plugged in during operation?
Most likely in consideration of an I2C'd 1602.
Could all of the constructors and setup be put into a function into loop or is there something special about putting them outside of loop and setup?
Example of goal: Arduino controlling LED strip, playing predefined patterns, LCD gets plugged into I2C, and it displays the name of current pattern without a reset.

Constructors usually do not communicate with a device. fmalpartida's i2c lcd library also doesn't if you use the below constructor

LiquidCrystal_I2C::LiquidCrystal_I2C(uint8_t lcd_Addr, uint8_t En, uint8_t Rw,
                                     uint8_t Rs, uint8_t d4, uint8_t d5,
                                     uint8_t d6, uint8_t d7 )

So you can keep that as is.

The ::begin does communicate with the device, so that needs to move into the loop().

You can test if it's connected in loop (based on Arduino Playground - I2cScanner)

    Wire.beginTransmission(lcdAddress);
    error = Wire.endTransmission();

    if(error == 0)
    {
      // begin
      ...
      ...
      // print
      ...
      ...
    }

I don't have an I2C lcd, so can not test. But it seems feasible to me.