Aren't these resignations done so that they don't have to participate in any 25th Amendment proceedings, whether so they personally wouldn't be forced to be in the position to choosing to do so, or somehow not having enough officials to carry out the procedure (admittedly I don't know the details of that or whether not having certain number of Cabinet members matters)
Finally had the time to do the research on this and it's not clear, but here's what I know:
Vice-President has to invoke the 25th Amendment along with a majority of Cabinet members. That poses two questions: how do vacant positions count and what about Cabinet members who have not been confirmed by the Senate? So assuming that can all get worked out in 12 days, likely via the Supreme Court...
The President can then ask Congress to intervene by submitting a letter to Congress arguing he is fit to serve in the role. Congress would then vote on the issue, but it would require a 2/3 majority of both the Senate and the House. There is very little chance this new House would approve by two-thirds.
So that leaves impeachment, which you already know would only require House majority (doable) and 2/3 Senate (questionable).