Ah, another "HUDSON HAWK" fan ... I really liked that movie because I looked at it exactly as you described: everyone having fun with a rotten script. I particularly liked James Coburn playing a take-off on "Our Man Flint".
And......David Caruso, playing himself. Athough this was prior to him wearing shades, posing dramatically, and whipping them off to emphasize every third line. (I call this "Carusitas". Not quite as good as "gravitas", but an attempt nonetheless.)
FWIW, cast list here