I’m am a first time visitor to this site and what I have seen so far is great, wish I’d found it a long time ago.
I am by no means an expert mechanic but would like to relay my recent experiences with tappet noises and the hydraulic clearance units. I have had my 83 500SE for about one year and have had a similar noise when the engine is worm, I have always assumed it to be a tired hydraulic unit and recently spent some time investigating the problem. Like you I used a stethoscope to locate the offending source of the noise, next step was to make a relatively simple special tool to compress the valve spring so that the rocker arm can be removed and then remove the hydraulic unit. I inspected the ware surfaces of the components and disassembled the hydraulic unit including the non return valve at the base of the piston, I then re-assembled the unit filling it with oil before installing the piston. This exorcise was performed on three valve sets ( one either side of the suspect source of the tapping noise ) results were as follows.
1. I used a clamp arrangement to bleed down the hydraulic units to facilitate re-assembly and found that the suspect unit took approximately half the force to compress ( oil was being displaced at a higher rate than the other units ).
2. I inspected the seating surface for the non return valve and could see a small amount of pitting ( if the non return valve passes the unit would require less force to bleed it down.
3. On close inspection of the rocker arm ware was evident on the camshaft contact face.

In conclusion, I will be replacing the rocker arm and the hydraulic unit for new items ( yet to be fitted ) I hope this is of interest and if you guys think I am barking up the wrong tree please let me know as new Merc parts here in Perth Australia are expensive.
I also have a request for information – What is the ignition timing settings for engine type 117963, it a Hong Kong import and I believe Euro spec car, vacuum advance only ( no vac retard ) have been told locally that it should be 16 BTDC plus or minus 3 with vacuum advance connected.

Good luck with the tapping noise

BobH