I imagine with breaks, lack of 100% focus and efficiency, and employment benefits it's probably $2-$3 in labor costs per test if you're right on the time required and average salary. So, I think possibly multiples of what you quoted, but still so low that it's not likely to affect the cost much.
Then again, with the people to move the samples around (process and sort mail), clean/sterilize testing materials after testing, and manage the people that perform the tests, and do quality testing, you might see another $1-$2 per test in labor costs. So maybe $5 in labor all said?
My bet is that the most expensive part of the process is the machines to do the tests, which likely cost multiples of a lab tech's salary (but maybe can do multiple tests at once? I dunno).
A sibling comment upthread noted that Maryland got 500k tests for $18 each, which sounds like it's probably close to cost, if it's the same test. That number of tests would be about full time work for 400 lab technicians for 5 days if they averaged just under two minutes a test.