how not to carry investment income to a non-resident state return

When you need to prepare a resident state return and a nonresident state return, prepare the nonresident state return first. Then, if you had to pay tax to the nonresident state (tax shown on a tax return, not withholding), your resident state normally provides a credit for taxes paid to the other state.

If you started preparing the Illinois resident return first, but haven't filed it, you can delete it and prepare it again after you complete the Wisconsin nonresident return. All of your taxable income should be taxable by Illinois since you are a resident.

When you complete the nonresident Wisconsin state return, answer all of the questions in the state interview section, so that you identify only the income that was taxable to Wisconsin. If the Wisconsin withholding was an error by your employer, then report that you had no Wisconsin income. Wisconsin only taxes nonresidents on income from Wisconsin sources. See this Wisconsin Department of Revenue webpage for more information.

Then, prepare the Illinois return again and do not answer that you had income taxed by another state unless you showed a tax liability on the Wisconsin return.

Some states have reciprocity agreements so that you normally don't have to prepare two returns if you only receive wages in the nonresident state. Illinois and Wisconsin have such an agreement. See this article for information on states with reciprocity agreements. In your situation, you need to file a Wisconsin return to get the withholding back.

See also this TurboTax article and this one on multiple-state situations.