Just put in a number. Then explain what you did in a written statement as part of the tax return. You may get a letter from IRS, but you reported all your income properly.
We have filed returns by snailmail for 10 years with no number and have never had a problem. E-filing doesn't allow for clarifying statements it just bounces the return, number or no number.
If I removed her income completely I could also make it work electronically but then I run the risk of a fine for underreporting income, even though her income barely makes a dent in our total household income.