The '55 and '56 'Birds were Dana 44s, CK. The picture you've posted does not look like my recollection of what the '55 and '56 'Bird rear end looks like. Limited slip versions of the Dana 44s were used in a number of different applications, but I'm rusty on what.
Wikipedia has a lengthy write-up on the Dana 44 with input from several writers. I've pasted a piece of it below. Always double check Wikipedia info, but it's a start...
Scanning the Wiki write-up, it indicates that there have been numerous variations of the axle assembly to include:
- solid front AND rear axle, independent front AND rear suspension
- ring gear size (8.5 and some 8.9 inch)
- axle shaft diameters and spline counts
- pinion shaft diameters and spline counts
- gear ratios and two different carriers depending upon what gear ratio you're running
If you look at eBay, and do a search for ‘Dana 44 locker’,and then ‘Dana 44 limited slip’ you’ll get numerous listings of new units available that are:
- clutch type
- air operated (!)
- electro-magnetic (!)
These are offered by Detroit Locker, Eaton, Auburn Gear, USA Standard Gear , Aussie Locker, Spartan, ARB, and Yukon. Most of these appear to be for 30 spline Jeep applications, but I only did a cursory look.
From Wikipedia:
The Dana/Spicer Model 44 is an automotive axle manufactured by Dana Corp. and is used extensively among automobile manufacturers and in the automotive aftermarket area as well. The Dana 44 was first manufactured in the 1940s and is still being manufactured today, both front and rear axle variants. The Dana 44 has been manufactured as a beam axle and independent suspension for both front and rear axle setups. There are also different variations of the Dana 44. Over a dozen automobile manufacturers have made vehicles that feature Dana 44 axles, including Jeep which currently manufactures Four-wheel drive vehicles that feature both front and rear Dana 44 axles.
General specifications
- Ring Gear measures 8.5 inches (216 mm)
- Ring Gear measures 8.9 inches (226 mm)
- OEM Inner axle shaft spline counts are 10, 19, 29, 30, 32, 33 and 35
- Pinion shaft diameter: 1.375" (most Dana 44's) 1.625" (JK rear 8.9" Dana 44's)
- Pinion shaft splines and spline diameter: 26/1.12" (most Dana 44's post 1970), 24/1.25" (JK), 10/1.12" (early pre 1970 Dana 44's)
- Gear ratios range from 2.72:1 to 5.89:1
- Carrier break: 2.72:1 – 3.73:1 and 3.92 – 5.89:1.
On some differentials only the high speed 2.72–3.73 carrier is used for all gear sets up to 5.89 (JK, Nissan or when thick gears are used)
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Hope this helps. 
NoShortcuts
a.k.a. Charlie Brown
near Syracuse, New York