The tests I named are all individually-administered. Yes, it is possible the current school will not be as interested in testing him, once they know you are re-locating, but I would still consider pursuing it with them, as plans can change. Plus, if they offer it to you, and you sign consent for him to be tested, they really aren't as likely to rescind the offer.
The kind of individual intellectual assessment that you might be considering usually costs between $200 and $600, depending on your region, the clinician, and what is included in the assessment. If additional assessments (academic achievement, social-emotional, behavioral, neuropsychological, etc.) are included, the price goes up from there.
There is no real "right age" for testing. It's more that you have to consider your purpose in testing, and the nature of specific children at various developmental points in their lives. Generally speaking, if you want a single measure of intelligence to be representative of lifetime ability, the preferred age is after age eight or so, when scores tend to me more stable. If you need a measure for access to resources (e.g., entry to a specialized program or school), then whatever age you need those resources is the right age to assess. If the purpose of testing is to access adult services (e.g., for accommodations for learning disabilities in college entrance exams or other post-secondary educational circumstances), then the right age is two-pronged: one test at least three years before the accommodations are needed (to establish that it is a long-standing documented disability), and one at or after age sixteen, so that it will be on an adult form of the test (if the earlier test did not have contiguous norms up into adulthood, such as the KABC or WISC), and so that the most recent testing is no more than two years old.
When searching for appropriate private assessment professionals, many people have had success with the lists on Hoagies, contacting your state gifted association, calling area gifted private schools for referrals, and asking acquaintances who have children in gifted programs.
