If your Android is rooted, it's pretty easy to get tethering working. There's magisk modules that can fix the TTL problem and/or disable the hidden carrier-installed software that Android will ask for permission before enabling tethering.
Aah, you mean ‘snitches’. :P
Going on a bit of a tangent, but deep packet inspection can identify packets routed using NAT, so if the phone is operating as a typical hotspot it would be identifiable by your carrier. Carriers in the USA used to block / denylist / charge extra for tethering using this exact approach.
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.
The delineation here is between "shallow" packet inspection (which basically nobody refers to because it's just a normal part of networking), where network devices look at just the bits of the packets they need to route / NAT / etc them appropriately.
DPI can tell a ton of things without needing to MITM encrypted layer 7 traffic.
A boring example is that you can tell TLS from OpenSSH traffic just by seeing the initial handshake. sslh ( https://github.com/yrutschle/sslh ) takes advantage of this on the server side to let you run both on the same port.
A less boring example is identifying OpenVPN, Wireguard, etc traffic regardless of what port they're run on, to enable blocking VPN traffic on a network.
You could spoof it by finding out your mobile's TTL, overriding the TTL in the connecting device to be one higher than the mobile.
Does the phone add a proxy header? Can it be configured to not add the header?
Android and Linux use 64 by default - the block could be circumvented by setting the laptop to use 65 TTL.