As H4 > your spouse can't work unless you:
1) Are the principal beneficiaries of an approved Form I-140, Immigrant Petition for Alien Worker; or
2) Have been granted H-1B status under sections 106(a) and (b) of the American Competitiveness in the Twenty-first Century Act of 2000 as amended by the 21st Century Department of Justice Appropriations Authorization Act. The Act permits H-1B nonimmigrants seeking lawful permanent residence to work and remain in the United States beyond the six-year limit on their H-1B status.
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However, one can apply for TPS regardless of current immigration status (as long as you are continuously present in US since 06/24/2015).
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with approved TPS (assuming one also applied for EAD), one can LEGALLY work regardless of immigration status.
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That being said, your spouse can apply for TPS. Get EAD. Start working.. File H4.
H4 and TPS will not create any issue in future. It's not ghammad.. make best of it.
