I would chime in here just a little because your situation is quite similar to what mine was in September last year (I arrived in the US end of September 2021, L1B visa with wife on L2). Wife got her EAD just this past month (July).
In my 2021 tax return, I was a nonresident alien because I did not meet the substantial presence test (which I assume you won't meet either given the time that you're arriving here). Therefore, my return was filed as "married filing separately" because my wife didn't get an SSN issued at that time either (the SSA is ridiculously backlogged and slow because of COVID). Knowing that that was going to be my filing status, that's what I indicated on my W-4 as well. Based on what the tax accountant that I consulted with told me, it didn't matter because a nonresident alien cannot claim deductions, only child tax credits. And so if that's the scenario that you think is going to play out for you as well, I hope knowing this will help.
Given our relatively similar situations, I'm happy to try and help answer any other questions that you might have as well. But yes, W-4 only affects your withholdings, not the actual tax bill which will come later when you file your return.
