I might be wrong about this, but I don’t see anything in that SkyLeader product that says it gives you position in real time. It appears to be a thing you download after the bird comes back and you can retrace the route it took - I think it’s just logging GPS location to internal memory for later analysis basically fitbit for birds.
One of the issues with GPS is that it is power hungry if you want the accuracy it is capable of. But if you only want occasional updates, and don’t mind significantly decreased positional accuracy, you can reduce the power draw markedly. There is a burgeoning industry for remote tracking, especially of things like shipping containers. Often they don’t need either fast updates or high precision. Usually they will take occasional quick and thus low precision GPS fixes, and then uplink to a LEO or Iridium sat. Nanosat startups are looking at similar ideas.
For existing tech, Yellowbrick trackers, and modern EPIRBs are some good examples of the basic ideas. Sat based AIS receiver services are another interesting technology and commercial market.
The thing about the OP’s preconditions is that they overstate the capability of government agencies. GPS is already a near cost no object system, created at enormous expense and effort. The cost of the current GPS satellite constellation alone (ignoring R&D and upkeep) is not that different the entire annual budget of the CIA. The CIA is never going to have the sort of money needed to create a super capable satellite system that does the such a niche job as asked for by the OP. Nor do government agencies have access to super-ultra tech that surpasses known existing science. Indeed for a lot of technology, once things get commercially successful, the consumer base becomes so large that the R&D exceeds anything a government agency could throw at it. There are only a few times in history that Manhattan Project scale efforts have happened. Things like high performance mixed signal integrated circuits are driven by the needs of the mobile phone market, and develop at a pace, and with budgets, that far exceed what the spooks cold reasonably muster. The spooks could (and probably do) commission the design of a bespoke chip for purpose, and might be able to consider a design that pushed the process to the absolute edge, with terrible yields, but it is hard to see them getting much further ahead.
So, as Chronos says, define the use case, and then look to what technology can be leveraged to get close.
Yes: define the mission, not the hardware.
As others have noted, specifying a transmitter sending signals to satellites pretty much requires power and antennas massively incompatible with “smallest”.
Well said.
DoD right now is in the middle of a massive rethink of procurement precisely because their current 1960’s-based paradigm is that their tech is far above that of the civilian space and faster moving.
Which has been a quaint fantasy for 20+ years now. Leaving them fielding machinery that’s ploddingly obsolete on its first day in service.
So they’re trying belatedly to jump on the “agile”, minimum viable product, feedback-driven continuous update model with dumb service busses and smart services. Whether we’re talking computers, communications, boats, trucks, guns, or airplanes.
Th extremely rigid laws surrounding procurement are a constraint, and the hidebound nature of both the DoD and the major defense suppliers like IBM, Lockheed Martin, etc., make this a very difficult transition.
But they are working the problem.
I’m not a government, so I don’t have an actual mission. I have designed systems to track large birds for thousands of miles years ago it was quite large. Argos is cool. What is the theoretical size for a transmitter within that system?
I’m a curious Engineer whose friend wrote a movie script.
I’m curious what small size is possible. Regardless of the other limitations. I have defined some basic requirements like distance. Ping rate and accuracy are not as important.
Well, right here
ARGOS says off-the shelf transmitters weigh 5 grams and transmit at 150mw. And appear to be about the 1x2x4 cm. In the photo of what looks to be a falcon the transmitter seems to include a solar array to extend the battery life, perhaps indefinitely.
Clearly something that needs to be exposed to the light to work is also exposed to being seen by whoever might be looking for it. Which is probably disqualifying for a clandestine device.
ARGOS trackers are also used for marine mammals. Though size isn’t a major issue there.
As always, there is a trade-off between size and performance.
ARGOS floats are scattered throughout the world ocean. Thousands of them. They spend 99% of their time slowly bobbing up and down in mid-water using buoyancy controls, and then daily rise to the surface, dump their data to ARGOS, and head back down for another data collection run. They run on batteries, but can carry enough to last for years. And slightly germane to the OP, depending on local currents and where they are dropped, they can travel anywhere in the world ocean, coastal ocean, or seas.
more relevant to the OP, consider microwave energized transponders. They are very small.
Here is what I am talking about. µTags : Passive microwave localization
They have been used to track butterflies.
As I understand it, people have built them to be powered by other microwave sources such as TV broadcasts and WI-FI networks. Probably cell networks as well. It would take a very good dedicated satellite to pick up the resultant signal through a ceiling but I suspect it would be possible. At least in principle. NASA can still hear Voyager signals for instance. If one was writing a book that is. 
This talk describes project Icarus, which is a satellite based bird tracking system that uses an antenna fixed to the International Space Station to track bird migrations. There are some interesting spin offs. Apparently the movement of animals can be used to predict earthquakes! They know something we do not. Clearly their sensors are better than ours.
I notice that trackers don’t use energy harvesting/scavenging techniques, they use a battery and small solar panels. I guess they need to enough power to send a signal to a satellite, which is quite remarkable. The current generation is 5grams, the next will be 1gram.