It would probably need to be a mixture of specialities for different tasks (Geologists, Biochemists, Astronomers etc.)
The next question is;
Who would start, organise and run the investigation? The chemist? The geologist? The astronomer?
Given the circumstances of an elusive UFO, a police investigator would be more qualified for this specific scenario imo.
Perhaps scientist studies and reports are initially not required to investigate this specific event (Unknown contacts flying around Earth).
It seems to be more of a criminal investigation, rather than a repeatable scientific study. I could go out on a limb and say, not one scientist is required to start this investigation.
If a crime is committed I don’t call a “scientist” (Botanist, chemist, geologist etc.) Science will obviously help the investigation.., but I see this only being a very small part of the overall task.
Its most likely that more money, time and effort will be spent on monitoring, and capturing an elusive extraterrestrial/man-made aircraft, rather than scientifically confirming the retrieved data (ET body/tech/trace elements etc.)
Although scientists may help in many ways.., they would not run the investigation, but would merely be given tasks.
1) A police investigator would be great at starting and running the investigation, and organising the facts. They work well with other agencies to solve problems.
2) The military’s technology would be great for detecting and pursuing potential criminals.
3) Scientific tests would be great for corroborating evidence.
Everyone has their strength and weaknesses, but a scientist just doesn’t have the skills required to do the majority of the work required.
Numbers 1 & 2 seem to be required for a proper investigation. But it seems as though scientists roles are very limited to running a few tests here and there to help police confirm radar, images etc. and simply confirming ET DNA.., after the DNA/evidence has been acquired, by the investigator.
If police/military shot down a UFO, and informed the public about small grey bodies being found within the wreckage of the technologically advanced aircraft.., then one could assume it may be an extraterrestrial.
However it could be some type of mutated Nth. Korean soldier, an affect from the crash, dummies or something else. THIS is when a scientist would step in to analyse DNA, trace elements etc.
They are not needed beforehand to start the investigation.
In short, get a bunch of police investigators from different towns, states, nations etc, to work on the case, like they would any other interstate/national crime.
Chuck them in an amateur astronomy course so they can tell the difference between a meteor and an alien spaceship if need be, but I’m sure they’d be capable.