Are there any identifying marks on the pen, cap, inside the pen?
Measure the diameter with your calipers on multiple threads to determine if the threads are tapered, and note the type of start, such as single start, triple start, etc. Then you would need to check the threads with a thread pitch gauge. You can do this with other methods as well if you do not have one, but they are typically not as accurate. Once you have your thread type, diameter, and thread pitch, use a reference guide (many online for free) and find the tap that meets your measurements. Try imperial and metric.
Here is a $10
thread pitch gauge from Amazon.
This is a good video showing the process: