You don't need the -5V supply at all. The INA122 will work down to 2.2V total supply voltage and its common mode input voltage includes ground. So ditch the 79L05 and connect pin 4 of the INA122 to ground instead. See Fig 5 on the datasheet.
Consider the situation where the unbalanced state of the bridge, Vunb, is <= 0.1V/G, where G is the gain of the instrumentation amp. While the common mode input voltage includes ground, the output voltage of the instrumentation amplifier when operated from a single supply does not.
The OP's original plan takes that into consideration.
That's a valid point. However, if a resistor of around 5K to 10K is connected between the amplifier output and ground, then the output voltage range will probably go right down to ground, unless the amplifier output stage is of an unusual design.