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.

You can, however you need to use market values of debt and equity. Book value of debt is good approx. of market value of debt. For equity, you can some multiple for that industry (EV/EBITDA, P/B…) to estimate market value of equity. Alternatively, if you think that company has the similar leverage you can use average debt/equity for that industry, but then the difference between benchmark beta and private company beta would be in the tax rates.
Again, what you CANNOT do is use book value of equity, because it’s meaningless number.