TurboTax does not support Form 843. Filing this form will not impact your tax return so you can e-file through TurboTax.
As you said, your social security earnings and withholding are consolidated on one W-2 so the IRS does not see this (on your Form 1040) as two jobs, which is why you cannot get a refund for excess social security on your tax return.
Form 843 and your tax return will be processed separately and any refund from the Form 843 will come as a separate check so waiting to file your personal income tax return or filing together will not make any difference in processing.
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