We travel with our dogs very frequently, including to the US & other international destinations. It is really no hassle. For the US, you usually just need a valid rabies certificate. Other countries may require additional paperwork if they allow dogs to travel there. Some airlines require a health certificate as well. For flying in cargo, temperature acclimatization letters are required by some airlines. Shih-Tzus & their crosses should not fly in cargo, however due to concerns about their breathing.
Your dog is right at the upper limit for in-cabin travel. Currently for Air Canada, the limit is 22lbs including the carrier. Could she go on a diet??
Our Shih-Tzu fits in the largest Sherpa Bag. She is the same height as our smaller (5kg, 7kg) dogs but due to her breed conformation, she is about 9kg. So in her carrier, she just squeaks under the 10kg limit.
But honestly, no airline has ever asked to weigh her. They ask over the phone at the time of booking, but that's about it. She can stand up & turn around in her carrier, so that is the most important thing.
I know you have decided not to fly with your dog, but I thought I would send this answer, in case anyone else is wondering...
Some dogs are just as happy to stay at home if there is someone there to feed them, walk them & rub their furry bellies!