What I would probably do is follow exactly the same schematic that you see in the LT6106 device.
You need to find an op-amp whos inputs can operate at the + supply rail, and an output that can operate up near the + rail voltage. There are a number from companies like TI, NS, Linear Technology, Maxim. Rail-to-rail opamps are fine.
Replace the transistor with a P channel small signal mosfet with a 100 ohm series resistor to the gate.
It is really important that the voltage on either input cannot under any circumstances get more than 0.3V above the positive rail, so if it is possible, you can add some 1K series resistors to the opamp inputs, and reverse biased schottky diode from the inputs to the + supply rail.
Now choosing the value of the series resistor is totally dependent on the offset voltage of the opamp. If you go for a low offset opamp with a 10uV offset, and you have a 3 1/2 digit meter, then you can quite happily work with 5mV drop across the sense resistor at full scale. If the maximum current is 5A, that would mean a 1mOhm sense resistor. You have to make sure that there is a 4 wire connection to the sense resistor - ie the opamp connection to the resistor does not use any of the PC tracks carrying the full current.
If your opamp has a 100uV maximum offset, then you want to go for something like 50mV voltage drop on the sense resistor at maximum current.
Calibration is easy - just add a pot in series with the resistor across the voltmeter input.
You will not use that many extra parts, you can choose a mainstream opamp instead of a single source device, and the accuracy and linearity will be much higher then the LT6101.
May even be cheaper.
Richard.