If a 1ms pulse is enough to change the state, do not change all the states at the same time. If you have 7*4=28 segments, driving them all sequentially at 1ms intervals would still allow changing all the segments within just 28ms, so the sequence would be just barely noticeable to human eye. If it needs to look instantaneous, maybe drive a few at once, still a significant reduction in peak current compared to driving them all within 1ms. Anything within 10ms looks instantaneous to most people.
This way, peak current is reduced and power supply is much easier to source, also voltage drops, inductive kicks from stray inductance and EMI are all reduced.