The existing light is atleast a little over 4ft so im assuming 3 joist from where the new lights will be mounted
But you still have to get above the ceiling to get the wire down to the switch, which will probably mean making a hole in the ceiling for sure, and maybe the wall if you can't get the cable through and existing hole because a drill with a bit in it might not fit above the ceiling to drill down.
Pull the switches out of the wall. The first thing to do is see if there is a set of neutrals (white) spliced together inside of each box. If there isn't, give up on the idea of fishing down to a switch. If there is a neutral passing through, your luck just got better. There will be two three ways and at least one four way. Forget about the four way, you need the switch leg end of the three ways. The three way switches will have three screws plus a ground terminal. Two of the screws will be the same color, those are the travelers, and the third screw will be a different color, usually black. Take the wire off of that third screw on each switch and using a meter, not a glow pen, check for voltage between that wire and the neutral. One location will have 120 volts present all the time, the other will have zero. The switch that is hot all the time is not the switch you want to use, you want the other one which will have the wire going to the light(s). If it's the switch you wanted to feed from, great. If not, it's time to make another plan.