3 Wire Load Cell, but whitout using Arduino

Using Arduino General Electronics
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use a multiturn potensiometer (or resistor network) to generate voltage for one input. This voltage should be equal to loadcell output at no load.

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The circuit you want is the classic Wheatstone bridge. A 3-wire sensor is just one arm of the bridge,
so you need a 2nd arm made of two precision resistors (and optional trimmer in the centre).

One of the arms goes to IN+, the other to IN-, they share the power connections.

Note the need for precision resistors, I'd recommend 1% or even 0.1% metal film types.

Perhaps two 1k metal film resistors and a 47 ohm trimmer in the centre, cermet type, like this:

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+PWR, -PWR

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The gain of your amplifier will need adjusting to a lower value if its too high...

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Connecting a large resistor between IN+ and IN- could help to reduce sensitivity. I would try a 100k but it is only a wild guess.