Profundis is profoundly funny as Tom struggles through his appointed role as Jesus, complete with versions of money changers, Pilate, and Mary Magdalene. He's baffled by the proceedings, and naturally has no idea what's really happening. There's a certain deus ex machina element to the story which adds its own level of humor, and which Cowper pulls off nicely
It's true that Cowper leaves quite a number of loose ends dangling. The ending is not quite in sync with the rest of the story. Tom passes in and out of character. But the whole is funny enough, and carried off well enough, that it doesn't really matter. It's not as strong a book as it could have been, but it's a fun, light read.
If you haven't encountered Cowper, this is as good a place as any to start. Profundis is not exactly exemplary, and far from his best, but it is enjoyable.