I wouldn't shoot it with the rod unsecured. The rod will likely loosen more during use and tie up the cylinder so that the cylinder can't be opened with the thumbpiece as usual.

Rod collars used to come in three different lengths, .150", .179", and .190", to accommodate changes made during fitting. Now, the current production standard rod collars are usually .175" -.180" , so you may be able to open/close the gauge if needed with a new collar, depending on how long yours is now.

I would check the OAL length of the center pin to confirm that it is properly fit to the current extractor rod. You can do this by tightening the rod using your fixture, then push the breech end of the cylinder assembly down on a flat surface to push the end of the center pin fully forward. When checked this way, the front of the pin should come to the end of the extractor rod, or just slightly (.004 - .010") past the end.

Also check the muzzle end of the center pin for "peening" damage. The pin can be damaged during recoil, and this "mushrooming" damage on the end can cause the pin to become trapped inside the rod. This ties ip the cylinder and makes the cylinder hard to open.

The other issue that comes to mind is what you said about "very little" endshake. It's possible that the cylinder may be binding on the yoke, or there may not be adequate gauge in the assembly for the cylinder to rotate as it should, particularly when the gun is in use. Some "endshake" or gauge is required in the assembly for the gun to open and close, and still accommodate shooting debris, the heat generated by shooting, and lubricants. (.002" or so) Yoke misalignment can also make this situation worse, so be sure the yoke is aligned properly.
You can check the amount of endshake by measuring the barrel/cylinder gap, and then holding the cylinder back and measuring again. The difference in the two measurments is your "endshake". Be sure to check in different chamber positions, as the gauge can be slightly different one or two chambers over.

If your front gauge is only .002", debris on the barrel extension and on the face of the cylinder may be hindering the proper rotation of the cylinder and contributing to the gun being hard to open. You need about .004" in terms of front gauge, particularly if you are shooting lead bullets. Atomized lead buildup can cause all kinds of issues if the front gauge is too small.

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Ret. LE, FA Instr, S&W Armorer