The original gauge on a TR2-3B was mechanical, not electrical. TR4 onwards used electrical versions. That panel on eBay has one of the early TR4 electrical temp gauges installed (which is visually almost identical to the TR3 gauge), and the gizmo mounted on the back is called a "voltage stabilizer".
Unfortunately, that is not enough information to know if your gauge needs a "stabilizer" or not. Smiths made both kinds, generally their aftermarket gauges used balanced movements that did not require a stabilizer, while their OEM gauges needed the stabilizer.
Since you already have the gauge and stabilizer, probably the easiest way to find out is to hook it up without a stabilizer and see how it performs. If the needle moves up and down slowly, then it's probably the kind that needs a stabilizer. If it moves quickly (falls instantly to the bottom peg when power is removed) then it probably doesn't need the stabilizer.
Of course then you need to check accuracy anyway. Not all senders are the same (and some vendors don't seem to recognize that fact), plus the gauges themselve sometimes lose calibration. The voltmeter from my Stag (which is the same basic movement) was so loose inside that just rapping it on the bench would change its calibration.