My son is 2 (27mo)
Hi there and welcome to the gifted board.
Testing at this age is unreliabe. Most won't consider testing a child under 4 and even then, it's reccomended to wait until around 7. You say there are programs he'd qualify for with scores but I suspect these are programs for children a little older. Go and find out what these programs are, when they start and what your child would need to qualify. There are several IQ tests available and schools can be picky about which ones they accept. You don't want to be stuck re-testing because a favored program doesn't accept the scores you can provide.
I know the qualifications for early enrollment sound ridiculous but the reasoning behind it is actually pretty sound. By wanting your child to be 2.5 years advanced
My eldest was similar at this age (except that she had lots of stranger anxiety). I agree with Turtletime. Take it a year at a time, or at this age even just a few months at a time. Don't worry about testing at this point. Don't borrow trouble.
Miranda in rural BC, Canadamom to three great kids and one great grown-upunschooler, violist, runner, doc
My 6-year-old was like this at 2, except he has fine motor issues so didn't like puzzles.