I guess that must be the “bonus mystery part”. I couldn’t figure it out either. It has threads on the open end but like you said doesn’t fit anything. It is intended to be inserted into a funnel to strain the paint into the gun cup and is not a part that fits onto the spray gun. I don’t know if HF has a paint funnel with threads to fit that strainer or not, but I think the tempera/Ti mixture would take forever to strain through. I just ran the sprayer at full blast and it worked well, I might try throttling it down a bit on the next batch. These last two tiles turned out nice and black. A couple of letters on the first one look faint, but I think it is the light, the tile looks great.
The U and A on the second one are a little faint, I think I accidently touched the tile right there or something.
Good luck