Is it unethical to supply a marked solution to a student who has shown intent to cheat, with the intent of identifying them for punishment?
[We] had a lengthy discussion about sting operations and entrapment


The distinction between the two is important here. A sting catches someone doing something they were already going to do. Entrapment is where you trick them into doing it in the first place. If you posted an advertisement offering to solve the assignment for them, and some students took you up on that, that would be entrapment, and would definitely be unethical: they might not have cheated if you hadn't posted the advertisement.

But in this case, they have already chosen to cheat. The only thing you can change is whether the assignment that they cheated on is easily recognizable as such. In some cases, it's hard to distinguish between a sting and entrapment. That isn't the case here; there's a (presumably) timestamped job posting the proves they were planning to do this before you got involved. 

You should still definitely run this by the department chair or someone first if you decide to do it, though. It looks similar to entrapment even though it isn't, and it's better to justify your actions before you do them, instead of letting it be an unpleasant surprise when the student complains later.

The fact that you'd be getting paid to do the assignment is an issue, as well. Returning the fee to the cheater after catching them probably addresses this issue. (And maybe turn it over to the department in the meantime, to avoid the appearance of potential impropriety.)

Edit: A comment requested an unambiguous conclusion, so:
My view is that it is ethical, but it's very close to the border, and either encouraging cheating that would not have otherwise happened or profiting from the actions you take would push it over to the unethical side. And because it's so close, it's important to have some amount of transparency (e.g. by telling the department chair in advance) so as to avoid both the potential for and appearance of impropriety.