Low current strain gauge
If you don't need to take measurements all the time you could put a P-FET on the  voltage going to the bridge and turn it on only when you need it.

The FET just needs to have low threshold voltage (Vgsth), also watch out for the Rds on resistance of the FET that may be relatively high.







simulate this circuit &ndash; Schematic created using CircuitLab

For example wake up every second (or five, or whatever you need), turn on the FET, take a measurement, turn off FET. This should be able to happen in much less than a second, so your average current becomes low. Or, turn on the FET, take your readings for 10 (or however many) seconds, turn FET off. The specifics depend on your application.