Congrats on your new boat! What did you get and where are you located (proud papa pics too.....)?
Part of the reason I ask beyond curiosity is it can make a difference in recommendation and advice. Would also be helpful to know size of the inverter and if it's wired to run the entire boat or just key components (though some heavy loads are usually not run through the inverter). Would also be helpful to have an idea of your intended cruise - how big is your family, where are you headed. Will you be anchoring or staying in marinas or both? Will you move every day (giving your batteries a chance to recharge) or sit at anchor for a day or two?
Your OP has two components. When to use an inverter, and battery capacity which will play a big role in how much you can use your inverter. For capacity, you have to consider the 12vdc loads too, not just the 120vac because they all share the same reservoir of energy - the 4D battery. Assuming a healthy battery, you don't want to discharge more than about 50% - which is 12.2v for a lead acid battery. You have around 75AH to work with (half the 150AH rating of a healthy 4D). Not a lot, but workable. It's surprising how much power it takes to run/charge small electronics such as tablets and phones. An old-school incandescent anchor light will consume up to 10-12 AHs overnight - 15% of your overall capacity.
At anchor, you can probably run the microwave or electric coffee maker for short periods off your setup. I say "probably" because with high amperage draws off a small battery bank, the voltage of the battery 'sags' and may trip a low voltage alarm in your inverter. With engine running, the alternator is providing enough power into the battery to offset the power being drawn from appliance. Or you may find that what worked fun in the day when battery voltage was high won't work well in the morning after 12-hours of voltage decline.
Many people develop a rhythm of generator running. A couple hours in the morning to charge the batteries, heat water for showers, and cook if they have electric range vs propane. They also run the the generator for an hour or two in the evening, perhaps coinciding with running air conditioning.
BTW - depending on your alternator setup (twin engine vs single, large output alternator), will probably take at least a couple hours of engine run time to fully recharge a 50% discharged battery.
Hope this helps. Probably more info than you need.....
Peter