There are no published backlash dimensions to my knowledge.
First you need to drive out the cross pin lock pin - it goes out away from the crown wheel and is locked back in place by a burr across the hole on the non crown wheel side of the carrier.
Check the cross pin for wear and renew as reqd. (pt no 100844)
Check the gear teeth for surface break up - renew as a set - gulp!!! The same diff gears are to be found in Triumph All TR3 Girling, TR4 TR4A TR250 TR5 TR6 Dolomite Sprint, Triumph 2000, 2500 2.5 PI, and Stag axles. Fit as a set tho!!! - Also the carriers are all the same except Stag.
You need to get the sun and planet wheels out to measure the existing washer thicknesses to give you an idea where to start when buying.
I start by fitting new washers behind the sun wheels (the big ones with the spline in them) then refit the planets with the old thrust to get a 'feel' Today you may find the replacement sun wheel washers are fibre not brass - they are in one size only.
Increase the size of the planet wheel washers to close the gap and reduce backlash as required using the dimensioned items available.
What you are after is a tight but free running movement when turned by a shaft in either side.
DO NOT FORGET THE AXLE SHAFT THRUST BUTTON WHEN YOU RE-ASSEMBLE.
Do not be afraid to reuse your original sun wheel washers (asumimng they are ok quality wise) to get the right running feel.
It is a game of trial and error.
The axle end float is critical to get the right clearance on the wheel bearings.
Not sure what the grey colour in the oil is.
Enjoy the day or so you will need to do this whole axle rebuild
Peter W
PS a hypoid case spreader is not required to extract the crown wheel carrier asy - I have always used 2 tyre levers.
Pinion bearing preload without the seal in is important - make up a spring balance to check the rotational load.
If in doubt folow the book (!)