Yes, Patrick conscriptus is correct, and your expression can indeed be read as, “the Patrick who was written/drawn up/composed.” And if your goal is a pun on patres conscripti (singular: pater conscriptus), then that's probably the best you can do. Usually I would at this point advocate for using the original Latin form of the name, Patricius, but in this particular case I admit it might ruin the wordplay.
However, “written” is not the most natural reading. If you talk about an actual written work – say, a liber, epistula, tabella or testamentum – then the verb conscribere is obviously most naturally understood in the sense of writing. But when talking about human beings, a more specialized sense prevails, as you have seen: It means to write somebody's name on a list; especially enrolling someone in the military (cf. the English term “conscript”), but also for other purposes. You can compare this to the English verb “write up.” If a report is written up, it's actually written. If a motorist is written up, he probably violated some traffic law.
Note that the term patres conscripti for the members of the Roman Senate is an example of this, although it's not clear who is on what kind of list there, and even in classical times people were already confused about the exact meaning of this traditional moniker. Many modern reference works agree that the patres and the conscripti were originally two classes of members, i.e., it should be thought of as patres et conscripti, but by the classical era, there was certainly no such distinction.
Given that you want to write a pun about an unusual situation – a person that was actually written – I think you can ignore the more natural reading, though.