the only reason the LT1920 is advertised as being an instrumentation amplifier is that it has excellent DC accuracy. this is due to the low input offset voltage, but more importantly the precision trimmed resistors inside the device. If the resistors are trimmed ratiometrically, the CMRR of an instrumentation amplifier is extremely high. If you can find an op amp with similar input offset voltage and strap precision resistors around it, you can beat the performance of the LT1920. The precision resistors will be expensive, but I see no reason why it cannot be done. Have a look at the LT5400. these look to have better precision than the resistors in the LT1920.
However, this will get you a precise feedback signal, but this signal is then compared with a (comparatively) imprecise reference voltage inside the chip, so you will loose the accuracy you have fought so hard to get. If you need precision CMRR, then your approach is correct. If you need precision output voltage, then you also need to look at the reference inside the chip as well as the feedback resistor accuracy
Best wishes from Analog Devices