Provided the tool is in good condition, and hasn’t been subject to dodgy repairs (some I’ve seen on eBay are in terrible condition or were incompetently repaired) the balance arms are clamped very securely when performing adjustments and the relevant surfaces are true.  This is an excellent tool in my experience.  Here’s a picture of a balance clamped securely, ready for an adjustment to be made : Best Regards, Mark

I don't use a lathe to reduce the diameter of the pin I use something else but I'm a bit puzzled as to why is it too big in this particular case is that's extremely big? By some chances as the balance bridge off another watch?

Yes, remove all the power first so that the pallet fork can be taken out safely after the balance. Fit the winding stem so you can wind the watch up, don't get into the habit of using a screwdriver to turn the ratchet wheel. While the delicate bits are out of the way, take note of any excess side and end shake in the train, including the barrel. Now wind in a little power and pay attention to how free the train spins. A good indication is when the escape wheel backs up for a second or two at the end. It doesn't necessarily mean that there is a problem if it doesn't do that. These are checks that you should do naturally during assembly. Saves time finding problems at this stage, rather thsn piecing it all together to take it apart again.

Any forger can add relevant dial print and case engraving, if he know what and where to put it.  This doesn't look real, but remove the back to find out for yourself Lily.

So I take the wound of the mainspring as I normally would do, before taking a watch apart. And looking if the escape wheel make some reverse motion?