The simple answer is maybe. Some devices that you run may be sensitive to over-voltages. You'll be equalizing at 15+ VDC. e.g., While the compressor motor on your refrigeration
probably wouldn't mind that, you may have electronic control boards that have components that might get damaged.
We have equalized occasionally in the past with our Lifeline batteries with the refrigeration on. I checked with SeaFrost (the manufacturer of our DC system, they said that "should be fine". It was.) One proviso: make sure you really don't go vay over the VDC that you're supposed to be charging at "a bunch". I wouldn't run our B&G instruments, even though they say they can take that over voltage, because I'm just a wimp; and, I'd avoid the radar being run.
On reason to avoid running systems: The equalization is based on WHAT IS ACTUALLY GETTING TO THE BATTERY TERMINALS. If you have a refrigeration running, you may be actually seeing lower voltage than you need at the battery terminals. Measure what's really being "seen" at the terminals. Our Mastervolt system and controls allows us to manually adjust the voltage so we can adjust it. Some other chargers don't do that.
I know a number of marine electricians who have recommended using an adjustable "standalone" charger. They actually monitor what's going on. I'd advise you not to simple engage the "equalization cycle" on your inverter, and then leave the boat on the expectation that nothing goes astray.
Are you just going to use your inverter on "equalize"? Make sure the battery bank is charged-up BEFORE you run the equalize function. You don't want to cooking the batteries (or you charger) by having it pumping a lot of amps into the bank. I believe you want to just bring the voltage up to equalize level without a lot of actual amperage being pumped in.
We are replacing our Lifelines with LiPro's because I had it with their tendency to sulfate. I'd tried everything to maintain them with a "religious" attention to not-discharging them with bringing them up to "full charge", equalizations every month or two, careful attention to temps of the batteries, and monitoring voltage at battery terminals during charging, etc., etc.
First set of Lifelines (on our current boat) from 19 years ago lasted 7 years. I mistakenly replaced them because "they owed us nothing" and out of an abundance of caution. Next coupe sets, lasted three or four years. Then two sets over
less than a couple of years each before their effective capacity became intolerable for 4- 4D's.
We have Lithium Ion - Relions being installed this month.
https://relionbattery.com/products/lithium/application/marine.
We have a MasterVolt charging system.