You should be able to make new purchases from either of two Apple IDs with such a set up, and possibly download them immediately. Downloading past purchases from either account is going to cause you problems however. Apple really want you to stick to one Apple ID per device. Family Sharing might provide a workaround.
Authorization vs. association
Up to 5 computers can be simultaneously authorized for each Apple ID. There is no direct record of which machines are authorized. It is up to you to keep track. (I use a small text file in my main iTunes folder.) iPods & iOS devices are limited to DRM protected content from up to 5 different accounts. There is no limit to the number of devices that you can load up with content using iTunes. You can deauthorize computers that you no longer wish to use to access protected content so that you can authorize others. You either do this on the specific machine or deauthorize all and then reauthorize as needed. Having used deathorize all you cannot use it again for a year, though Apple Support may be able to perform a reset if you can explain why you need to reuse the feature before it becomes available to you. Reinstalling Windows without deauthorizing first will use up an authorization.
HT201251: Authorize your computer in iTunes - Apple Support
A mix of up to 10 computers and iOS devices (no more than 5 computers) can be associated with your Apple ID for iTunes Match/Apple Music, automatic downloads, or redownloads of past purchases. There is a 90 day limit before a computer or device can be switched to another ID or the association removed. The list of associated devices is the one you can see within your account details in iTunes. You associate a device or computer by using certain features, typically downloading past purchases. You don't necessarily need to associate a computer or device with your Apple ID to use it with your content. Again Apple Support may be able to reset if you can explain why you need to reset the association early.
HT204074: View and remove your associated devices in iTunes - Apple Support
You can happily remove any and all associations that are not within the 90 day limit. Those that you actually use will be recreated as and when you make use of a service that triggers association.