I'm definitely going to be investing a good portion of my earnings back into the side business

This doesn't make a difference in how much you should withhold. What you do with your W-2 job income doesn't mean it isn't going to be taxed.

If your W-4 suggests that your W-2 job income means that you can use 3 allowances, that is just considering the tax liability related to your W-2 job.

For your side business, you can either send in estimated taxes that will cover the additional liability created by this business (based on the profit after you subtract your business expenses) ... or you can adjust your W-4 at work to have more taken out in workplace to account for that. Either approach you take is likely going to require you to estimate your tax liability. I'd recommend using Form 1040-ES for that purpose.

If you expect to have a profit, then you will have additional tax to pay (SE tax and income tax) on that profit, so "3" allowances might not be right if you decide you want to use the W-4 method. You would bring that down, not artificially inflate it, in order to have the correct amount withheld.

However, if you got a $9000 refund from federal and state last year, that suggests that your allowances at the W-2 job are not correct. Too much tax is being taken out during the year, so you should be using a larger allowances number.

You'll need to "do the math" using 1040-ES to figure this out. The IRS Withholding Calculator isn't good about handling side business work, since it doesn't account for SE taxes. The 1040-ES has you figure out your total adjusted gross income (from w-2 job and side business) and so it provides a better picture of your taxes.