@taxsavvy , I think I understand what you are trying to achieve. However , I am not quite sanguine about being able to do this. Here is a link you may want to familiarize yourself with -- > https://www.irs.gov/pub/irs-pdf/p503.pdf.
However, if you do decide to proceed with this ---
(a) Prepare your return including form 2441, claiming your mother as a dependent, fill out the details like name address etc. etc. In place of TIN, enter a dummy SSN, like your own SSN and changing the last four to 5000 for example -- I am assuming you are using the downloaded version ( because I am not familiar with the checks the on-line version does.
(b) Once you are done tell TurboTax , you are going to file by mail. Print forms required for filing
(c) Now go back and replace the dummy SSN for your mother with " N R A "
(d) Now Prepare a form W-7 including all the documents asked for ( copies of passport pages etc. etc. ).
(e) Sign and date everything as required and then file by mail ( please use tracking so you know it has been delivered ).
This would result in the return being directed to the ITIN unit ( I think it is in Austin TX ) and when approved the return will be processed ---- it will take quite a bit of time for your refund. In this case and if the ITIN is approved , it is approved retroactively for the tax year 2022.
The other option is to wait and file as and when her GreenCard application is approved ( but not after October 16th of 2023 ) and she has an SSN.
Note that , because she cam in on B-2 visitor visa her stay can only be extended to one year total and this may or may not be enough for the situation to settle down.
Good Luck on this effort ( at least Green Card )
Namaste ji
pk
