Sounds like - if you believe Andreas, a decent 1 mF aluminium electrolytic across the 3.3 V terminal and ground would make the choice of regulator moot and any one as good as any other.
I don't know if Andreas is right, but his scope tracings look pretty convincing to me.
For the project I'm working on, power will come from either an 18650 or three alkaline AAs, through a P-channel mosfet functioning as a power switch, to the 5V pin of the D1 Mini, then through the Mini's regulator to 3.3V. So I was just hoping to reduce the number of things that might contribute to "sagging" during transmission. The mosfet I was going to use (LP0701) has a couple ohms of Rds(on), so I think I will go with something better than that. I saw the IRLML6402 mentioned elsewhere, and it looks pretty good.
If I have to order a capacitor, I assume a 6.3V rating would be better than something higher - less resistance, more responsive, and of course smaller.
This is a mailbox notifier thing for a friend, and unfortunately the mailbox is not only metal but also enclosed in a brick column. I think there is some (small) chance the wifi will work through all that, but if not, we'll have to look at an antenna of some kind, or moving the circuit outside. Anyway, I just wanted to give it the best chance of working through the brick, which means keeping the power steady during transmits.