Quick update: As of 2.4 the attack speed/rare tome bug still appears to be in effect (timed them, I got 11 shots off in 15.84 seconds with 3-way staff and no speed upgrades, 15.63 with 1 upgrade, and 15.62 seconds with 2 upgrades) Was just using a stop watch so the differences are likely due to my reaction time.
Also, have not fought it today, but bleed was REALLY OP on the splitting boss yesterday, as the bleed was inherited when it split. After the first split I was usually beating the boss in about 5-10 seconds when I had Bleeding. Seems like the bleed should at LEAST be divided among them if not reset completely upon splitting.