OpenSecretes has a list of some Qatari PR contributions during their recent conflict with the Saudi-led part of the Arab world.
OpenSecretes' articles seem to be based mostly on FARA disclosures though. They don't seem to include less clear-cut cases like the AMI advertorial for Saudi Arabia, for instance. A more complete list like that is probably much harder to put together. I also don't know if the now closed pro-Saudi Arabia Foundation think tank did or didn't register as a foreign agent, for example.
There are some bipartisan US efforts to go beyond FARA, e.g. the Honest Ads Act, but despite being proposed some years ago, they have yet to make it all the way through Congress.