Using a 120W inverter with a 180W brick
There are a lot of interesting things in here, so I'll try my best to hit them all.

1) Just because the laptop power supply is rated at 180W doesn't mean the laptop will draw 180W.  If the laptop starts off fully charged, its actual power draw should be quite a bit less (after all, you can use the laptop and charge it simultaneously, right?)  So you should measure how much power the laptop ACTUALLY draws.  You can also adjust settings on the laptop to minimize power draw (laptops have power save modes, but you'll need to specifically turn them on since the laptop thinks it's plugged in and thus doesn't care about power usage)

2) If you do try to draw a full 180W from an inverter rated to 120W, you're not going to get 120W.  As mentioned in the comments, you will either blow a fuse or brown out the circuit.  That's why you need to know the true power draw of the laptop.  As long as the laptop never draws more than 120W, you should be able to plug in using the inverter just fine.

3) Getting a higher-rated inverter isn't likely to help you because the pilot has probably already sized the 120W inverter based on the capacity of his 12V plug.  But, doesn't hurt to ask.

4) I'm not going to go into much detail on the motorcycle battery since I think in your case the right answer is to run off the 120W aircraft power.  But if you do go the battery route, I strongly recommend measuring its lifetime at the actual power draw of your product + laptop.  As SamGibson pointed out (see comments), listed battery capacities are generally overly optimistic in realistic operating conditions.  I'm assuming flight time is expensive, so it's worth the time on the ground to figure out exactly how much operating time you'll have.  The battery datasheet should give you lifetime vs discharge current, but that will change depending on how old the battery is and other factors - the best bet is to measure it yourself.