Is BJT amplifier a viable option if the design needs to be replicated without calibration?
You don't specify the voltage across the shunt resistor or the accuracy required, but anyway the gain can be stabilized by use of negative feedback such as emitter degeneration. For this to work you need to design the circuit to have excessive gain and remove the excess with negative feedback. For example, if you need a gain of 100 you might design a circuit capable of a gain of 10,000 typically and apply negative feedback to reduce it to 100, so even if the gain varies from 5,000 to 20,000 because of the transistors, the (closed loop) circuit gain will vary less than +/-1%. This will also stabilize the circuit against temperature changes, since resistors are generally very stable compared to transistor beta. 

That may be all that is required if it is an AC circuit. If it's a DC circuit then offset voltage means that an op-amp is going to be better- you'd need differential transistors (perhaps matched and thermally coupled) and a complex circuit, and it still would not be very good.  

Op-amps (at the lowest end) need not cost much more than a BJT. 1-2 cents each in a quad package. Probably cheaper than a MOSFET.