Yes indeed you can run your AC from lithium batteries. Just not for very long. Underway you might be able to run a bit longer if the engine has a big enough alternator and it is configured properly to charge the lithium batteries at its full capacity without damage to the alternator. Note that you can run your AC from lead acid batteries if you have enough of them. There are several advantages to lithium batteries over traditional lead acid batteries.
First they come in at about 35% of the weight for the same amp-hr rating.
Second, and this is the big one, you can draw a lithium iron phosphate battery all the way down to zero without significantly damaging the battery. That is a big contrast to lead acid batteries where drawing the batteries down much below 50% of rated capacity on a regular basis significantly shortens battery life. Running a lead acid battery down to zero may effectively kill it.
Third, lithium batteries maintain voltage at near 13 volts for about 85% of total discharge. Furthermore they don't suffer anywhere as much voltage sag at high discharge rates as do lead acid batteries. This is good for your electronics.
Now the down sides.
First, lithium batteries can not put out really high currents for more than a few seconds. Most Lithium batteries are limited to 1C maximum discharge rate where C is the amp-hour rating of the battery. In other words a 100 amp-hr rated lithium battery will not put out more than 100 amps for more than a few seconds before its internal electronics (battery management system) turn the battery off. So high draw loads (windlass, thrusters, engine starting, windlass, etc.) can only be managed with a sufficient number of lithium batteries in parallel (think 4-6 or more for engine starting). A consequence of this is that you will need to maintain lead acid starting and thruster batteries with a separate charging system from any lithium batteries.
Second, lithium batteries have much lower internal resistance than do lead acid batteries. The consequence of that is that when the lithium batteries are discharged, they will draw a lot of power from the charging system. For example, if you have two (2) 100 amp-hr lithium batteries in parallel, they may draw up to 200 amps from the charge source. What this does is stress alternators by pulling as many amps from the alternator can supply from it. That will cause the alternator to run hot and can destroy the alternator, particularly at lower engine RPM. As a consequence, a lithium battery system requires an alternator with a sophisticated (external) regulator to prevent alternator over heating or an isolation system such as a DC to DC charger to limit the current that can be drawn from the alternator to a safe level. Furthermore lithium batteries require different charging profiles than do lead acid batteries, so the alternator's regulator needs to be capable of being set to the lithium profile. Also, due to the different charging profile of lithium batteries, you may need to change your shore power charger to one that has a lithium battery profile, or change the charging profile of your existing charger if possible. All these changes require the influx of MONEY.
Third, and this can be a big negative, there is a perception that lithium batteries can spontaneously catch fire and that it can be almost impossible to put the fire out. This issue is almost exclusively a problem for the lithium battery chemistry used in electric vehicle batteries. Marine lithium batteries are a different chemistry (lithium iron phosphate) where this problem is minimal. However, many insurance companies have a very negative view of lithium batteries because of this perception. As a consequence the insurance companies will likely require rather stringent installation requirements (compliant with ABYC standards) and also require that the batteries be UL listed. Most consumer grade lithium batteries are not UL listed. The cells in the battery may be UL listed, but the batteries a a whole unit generally are not. There are some fully UL listed batteries available, but at considerably higher cost. Again apply money.
The result is that lithium batteries, when properly installed are a good option for a house bank to increase usable capacity from the same battery foot print. Basically by switching to lithium batteries for the house bank you will effectively double the usable battery capacity. Lithium batteries are not a simple drop in replacement for lead acid batteries for the reasons outlined above. Yes you can run your AC from a lithium battery bank, but you will need a lot of batteries and after an AC run you will need a way to recharge the batteries.