I get a 'Sign in to a team account' error when I try to add my business account to the desktop app.

Welcome back to the community TGS2point0​ 

First, what type of business plan are you on specifically? This could affect what you can and can't link. More on that here.

But assuming you have the correct plan:

While signed into the business account

1. Log in to dropbox.com 
2. Click your avatar (photo or initials) in the bottom-left corner.
3. Click Settings.
4. Click the General tab.
5. Click Link accounts next to Link your personal account or Link your team account.

If you encounter an error, take a screenshot and share it here.

It's not a business account. We use a personal account for the business with one log-in / user. I spent an hour on chat trying for a work around. And even if your suggestion were to work, my second computer does not have enough hard drive space to sync all of the files I would need to access currently in my Dropbox.

Understood,

However, the second computer may not necessarily need to sync everything onto its hard drive at one time, depending on the circumstances.

But lets back up for a second, can you share what your ultimate goal is beyond linking the accounts? With that information, we may be able to explore some solutions. 

I need a solution where I can work remote at a moments notice with seamless access to 300gb+ of files and syncs them automatically.

TGS2point0​ You should be able to do this with the one account that has all of the files in it. By downloading the mobile app or signing into the dropbox account from another location, you'll be able to access the files or add new ones for syncing.

Can you help me understand the need for the second linked account?

It's not really needed. It's just there and probably not going away. It would be nice to be able to toggle from one to the other, but the workaround of log-off / log-on is workable. Storage is the main issue.