my understanding is that Meyer still uses only IIR filtering on its products
I was at a Meyer training last week and can confirm this, they do not use FIR filtering in the galaxy (their live processing unit). They do use FIR filtering in their Bluehorn system to achieve a flat phase response, however, that also incurs 45ms of delay which can be afforded as that is not intended for a live situation.
To my understanding, FIR will add delay, whether you can afford that is up to you.
In general, however, you will NOT hear phase shift in the context of the live domain, if you are curious try placing an all-pass filter on a speaker. Turn it on and off to see if you can hear it (I cannot). The only place I would be worried about inducing more phase shift is a crossover point or between two systems you are trying to use relative to one another. To the point any normative EQ work (i.e. no pass filters) should introduce a nominal amount of phase shift. If I remember correctly as long as you are in within 50 degrees you will not incur more than 1dB of penalty, which is more than acceptable in the live domain.
In general, the best system tuners I know of use very few EQ filters, generally between 1-4, and they usually have a very wide Q. Be careful not to overdo things. EQ is a blunt force instrument and there are many problems we cannot fix with it.