If the bead is seated, no need to pump up again.Just pull the core and add sealant.Then replace the core,spin the tire and pump. If the tire isn’t fully seated, dont insert the core.Pump the tire up without the core so yo are getting as much air into the tire as quicly as possible. If its holding air with the pump head attached to the stem, you should good to go. Then you just remove the head to let the air out so you can add sealant. Generally its a good idea install the core only after you have gotten the bead to seat both because you get air into the tire more rapidly and it reduces chance you will damage the core.
Thanks for your help. All this core in, core out confusion by me is because up until this point I had been unsuccessful. Thanks to the Airshot and my old trusty Zefal Husky floor pump, I think I’m finally understanding the workflow to be successful without so much trial and error. The Airshot has a little thread in connector that fits in the Presta valve minus the core. This set-up did indeed seem to let more air in, quicker the result being that I could hear more definitive popping as the bead was snapping onto the rim shelf around the full perimeter.
Today I plan to do the final step and add the Stan’s sealant. Yesterday I held off on adding sealant as I tested to see if the tire held air for any amount of time sans sealant. It really didn’t - it went from 81 psi down to 30 psi in 3 hours.