IMHO, the 2.5A official supply is getting a bit iffy with a 3B**+** and Pi Foundation display etc. I also have an analog audio amp and few other odds and ends attached to my display and 3B+. I usually run the 10A power pack as this is my bread board rig. I could have motors and or servos etc wired up. The 4A power pack I bought for another project and tested it to see if it was enough when the 3A wasn’t. Just wanted to know if 4A was enough for future reference. I only used the 3A because it was handy at the time. Won’t be using it for my bread board this Pi any more though.
The other reason I went with the barrel connector on that project is IMHO, its more robust. Only one way to plug it in, and its easy to plug in. No having to look to see if its right side up or upside down, or mangling it if you get it wrong.
Yes, the 4A should be enough, the 10A is overkill to be honest, even for what I use it for. The other way to do it is to run your Pi on one power supply and the display on a second one. “Two” official 2.5A supplies. Just throwing it out there. If you have two supplies on hand can’t hurt to try it. Don’t link the display to the Pi with the included power jumpers. Only connect the DSI cable.
I’ve run the 4A on mine a couple of times and didn’t notice any lightning bolt? Thinking about it, I may not have tried it since swapping the 3B for a 3B+? My 10A Power Pack is usually sitting on the shelf right next to that Pi. Sucks to have to make a second purchase. I have 2 of the 4A ones, one runs my Pirate Radio and the other I use to recharge the battery in, and run my portable weather clock.
Anyway, at least your up and running with no warnings. Now you can have some fun actually using it. =)