HS qualification is determined as of January 1 of the tax year, so she can only qualify for 2021. Check your deed of trust language but what if you granted your daughter a life estate? She'd be the 100% owner at that point and would qualify for the full exemption. You do not have to record a deed for it to be effective between the grantor/grantee and I have even seen some HS exemptions get approved by TCAD when the deed wasn't acknowledged and filed with an effective date until months after 1/1.
A timely application for the HS exemption is before May 1, but you can file up to two years past the delinquency date, which would be 2/1/23, and still get the exemption. Seems like they will have their financing straightened out way before then so the unfiled life estate grant, should get them the full HS exemption for 2021. Of course it depends on the exemptions clerk that is working the paperwork and none of this is legal advice.