Can I use the unlevered Beta to calculate WACC?

Here is the case, I am trying to value a private company and it is difficult to use regression.

So I am using the Beta of a public company and unlever it.
(i.e., Benchmark Beta / (1+benchmark debt/benchmark equity)

I then use the calculated Beta above and apply it to the private company to get Beta.
(i.e., Calculated Beta * (1+Private Company Debt/Private Company Equity)

Can I use this as the Beta for a private company?

If you are following by the book it should be:

Unlevered/Asset ~beta = \frac{Benchmark ~beta}{1 + \frac{Benchmark ~debt \times (1 - Benchmark ~tax ~rate)}{Benchmark ~equity}}

Levered/Equity ~beta = Unlevered ~beta \times (1 + \frac{Private ~company ~debt \times (1 - Private ~company ~tax ~rate)}{Private ~company ~equity})

If you do not have information on the tax rates, then a close approximation is to assume tax rates = 0, which are the formulae you had in your post.