DPI can refer to inspecting beyond just the headers, but since it's more of a marketing term than a technical one, you could also say you're "deeply inspecting" the IP headers of a packet and no-one would show up to arrest you for bad terminology.

Anyway, one way to detect NAT is to observe different TTLs originating from one device. Is that deep inspection? Probably depends on who you ask. The fact that you have to track information across multiple packets counts for something, though.

Off the top of my head I wouldn't really expect there to be much value in a MITM inspection of the contents of HTTP traffic for the purposes of NAT detection. You could probably come up with some scenarios in which it might be possible, but I'd content those scenarios aren't very practical. Easier to compare TTLs between packets, say, or track connections to known OS "phone home" destinations. While these just use information from the IP layer, they're stateful observations requiring comparisons across multiple packets, and that might count for something.

One way to detect a shitty carrier service, though, is that they're inspecting your traffic for "good" or "bad" uses of their service, because that is a good indicator that they're not just a carrier. I call it Dickish Practices Identification, or DPI.