Would there be a chance that they might mistake it for a tattoo done that morning, or the previous day, or something along those lines, or would they be able to definitively determine that the tattoo was done post-mortem?
They should be able to.
A tattoo is essentially a wound, or a million tiny wounds.
They can tell if a wound was made pre or post mortem, I don't see how the presence of ink in the wound would make a difference.
Tattoo a live body and things happen, swelling, redness, even while the tattoo is being done it swells along the lines. It bleeds.
Then very quickly it does what wounds do and starts to heal. It scabs over.
So a pre death tat should either show evidence of blood, swelling, redness or scabbing/healing, depending on how long before death the tat was done.
A post death tat would show none of these.
It could not be mistaken for an old, healed tat. There is an enormous visual difference between a new and a healed tat. Anyone could see that difference.
Also, I don't know if you have ever done tattoos (I have), you need a hell of a lot of stuff, even if you don't care about the hygiene aspect.
For a start, how are you running the gun?
You have two options, either you use a power pack that you plug into a socket, or you connect it to a car battery.
Is your killer carrying either of these around?
Then he needs a prepped gun, possibly two, or at least two prepped needle set ups.
One for outlining/detail work, and another for colour.
If he has two guns, one can be set up for each. If not he will have to change the needle set up in the single gun.
Then he needs his inks, for a realistic coin you'll need about four colours. So you need the stand for the caps, the caps themselves, and the bottles of colour.
He will have to fill the caps from the bottles in situ.
He cannot really load the gun from the bottle, and he cannot carry filled caps.
He also needs wipes of some sort to clear the excess ink from the skin so that he can see what he's doing.
He will also need some container of cleaning fluid to rinse the gun when he changes colours.
Then he has to set up, plug in, and set to work with his different colours. This will take a while, probably 20 minutes total.
Then he has to clear it all up, pack it away and get the hell out of there, with his kit.