Thanks for the quick response, @Vovchandr and @sltous !
The engine is indeed a 2.3L Ford Duratec! More specifically a 2003 1G271AA, which seems associated with the Ford Ranger. After you suggested I take a closer look, I managed to get a view of the side of the block and the stamping made it unambiguous. Thanks for the help!
Yes, the car was originally sold via Hillbank (this is the ad https://www.hillbankusa.com/vehicles/329/2003-caterham-seven-clubsport). That ad says 1.8L Zetec, and that's what the previous owner had told me. I just started using the car last week, so hadn't thought to second-guess that.
The previous owner was not much of a car guy, and he had a habit of driving without any filters on the ITBs. I am presuming there are lots of carbonized bugs and vegetation in the cylinders and that's one of the reasons it's running rough. However, I must admit that I didn't put the air filter on when going up to Alice's this weekend... I figured that one more trip wouldn't do much damage, since I'll have to have work done anyway.
Thanks also for the pointers to TrackSpec and Kampena. I had found TrackSpec already, but Kampena hadn't come up in my Googling.