FHSA qualified withdrawal to RRSP question

I have maybe a silly question but no one at TD seems to know what they are doing.

Below is a summary of what I'm trying to do:

• I signed an APS for a condo, closing on Jan 12th:
• I already have $16k in my FHSA
• on Jan 1 I want to contribute an additional $8k to my FHSA (bringing total to $24k)
• I then want to make a qualified withdrawal of the entirety of my FHSA on Jan 2nd
• I have funds to close my condo on Jan 12 without that $24k, so my question is - can I reinvest that $24k into my RRSP?

Is this all above board to get the tax deductions? I understand that the CRA does not trace FHSA $ as long as you have a qualifying home purchase, however is there any problem with withdrawing it and then depositing it into my RRSP directly?