Managed to get it on 4 toons total (3 alts, one main) on one Bnet account, only tried for 3 alts, but 4 might be possible without an authenticator. I disabled all addons, changed to minimum settings, set as windowed at the minimum resolution. I also have an SSD so that probably helps tremendously.
I opened 4 different clients and entered my password ready on 3 of those and used the fourth to log onto the character I wanted harrison on the least, since he's the least likely to get it. When I was ready, I logged onto the second client, entered my authenticator pin, then onto the next alt and repeated until I got to my main where I handed in the quest.
My guild mate was fairly confused when he saw three people all getting harrison at the same time.