I have a bottle rocket/pop bottle that I made a while ago and always used to just use it as standard.
Last week I was airing up a set of Dampf Evo’s on some old Sun SOS rims fitted with a Joe’s rimstrip and just for kicks tried track pump alone. It wasn’t looking like it would do it until I pressed down on the top of the tyre over where the valve is, and then they went straight up and stayed up. I’ve heard this technique mentioned a few times but it’s the first time I’ve tried it.
OP, I think you’re missing something about valve cores. The point of taking the core out is that it lets you get air in faster than with the core in, which is helpful for seating the beads. When the beads seat you’ll probably get a loudish ‘bang!’ from the tyre – possibly more than once. Go round the inflated tyre while it’s still on the pump and satisfy yourself that the bead is properly seated (heels of hands on edge of tyre to see if it comes away from rim). When you’re happy, hold the rim off the ground while taking the pump off. The tyre will deflate almost instantly, and if under it’s own weight against the wall could unseat the bead a little hence weight relief. Put the sealant in the tyre now, then put the core back in, then reinflate as normal.
Personally I like to add an extra step in and after seating the bead, put valve back in and pump back up to 30-40 PSI and leave the tyre like that overnight (valve closed, off the pump) to check it’s ok – then put the sealant in the next day.
Once the sealant is in, remember to spin the wheel to make sure it’s all over the inside, and don’t be alarmed if you see some little white spots appearing on the sidewalls depending what you’re mounting…