From a psych background perspective:
I'd be very surprised if you could find a psych to run an IQ test in a standard session. There is a lot of work that goes into completing an assessment, from initial rapport building with the child, collecting developmental history from parents, doing the assessment itself (administration + scoring/interpretation) and then writing a report and meeting with parents again to feed back the results. It's quite a time intensive task, hence the price tag attached. It is possible to have an IQ test completed at 3; however the results are often more reliable once the child is older, 6 years plus.
Given that the test itself is quite time and cost intensive, I'd question whether you really need to get it done at all. I personally wouldn't bother putting a child through it unless the information was really needed. At three, the value in having the information is really quite limited, as PPs have said.
As others have said, just provide your child with as many varied learning activities as you can for now. Once they get to school, you can talk to your child's teacher about extension activities if they need it. You'll find that schools have their own entry criteria for gifted and talented programs that usually depend on in-house testing results and/or teacher recommendations, and formal IQ results are not required.