Electronic – arduino – how to connect strain gauge to HX711 module

Are you sure the dummy strain gauges don't see any strain? You say it works with three 120\$\Omega\$ resistors instead of the dummy strain gauges. If the dummies don't see any strain, why are you using them in the first place? In a Wheatstone bridge you would either have one variable element combined with fixed resistors, or 2 or 4 strain gauges which work opposite to each other, so that the measured signal is doubled.

That horrible thing that you call a schematic looks OK, and your gain of 6000 should probably be enough to get some signal. Note that at a gain of 6000 less than a mV difference in the InAmp's input is enough to send your output all the way to ground or \$V_{DD}\$. You might need less gain.

I would try again with the 120\$\Omega\$ resistors.

Your first approach is unlikely to work very well because analog multiplexers don't exhibit good resistance matching between channels - in the case of HEF4067B, it's listed as up to 25 ohms mismatch. They also exhibit variable on resistance over voltage:

Your second approach should work because current won't be flowing through the MUX, meaning the mismatched resistance won't have much effect on the output reading. It may still result in offset errors that you need to correct per-gauge, but you'll have to calibrate them independently.

Another option would be to look for an analog frontend suitable to your application. There exist precision DACs with differential input, programmable gain, and multiplexed frontends, such as ADS1243 from TI, which would allow you to implement your entire front-end - mux, amplify and convert - in a single IC.