Does a battery's remaining power drop consistently regardless of remaining charge?
First, in answer to the first question: No, a batteries remaining power does not drop consistently regardless of remaining charge.

The simplest answer to the rest of the question is: you want to run the power inverter for as little time as possible. The explanation of this and method to accomplish it follows.

The power inverter uses some power itself, so you'll want to have it run for as little time as possible. This can be accomplished by only charging the battery up to around 60%-70% and turning off the inverter. The full step by step instructions might look like this:
1-turn on the power inverter with everything plugged in or ready to plug into the laptop
2-charge the battery to around 60%-70%
3-turn off the inverter, running the laptop off its battery until it's quite low again
4-repeat until the deep cycle battery and laptop battery are both empty.

Charging the first 60% to 80% of a battery can happen much faster than finishing the last 20% to 40% of the charge. You can easily see this by experimenting with your cell phone or laptop on a near empty charge; just plug it in at about 5%-15% charge and see how much it's charged at 15 minutes 30 minutes and 45 minutes of charging.

Here are some resources which explain some of these principles in action:

Chart showing how much charge is kept in an unused battery after 6 months -- ( http://batteryuniversity.com/learn/article/how_to_store_batteries )

"Ultra-fast charging only applies during the first charge phase. The charge current must be lowered when the 70 percent state-of-charge threshold is reached." -- ( batteryuniversity.com/learn/article/ultra_fast_chargers )

"As you can see from this one, if your inverter regularly operates below 20% of its rated power then your efficiency is down the toilet!" -- ( http://www.solarquotes.com.au/blog/two-reasons-you-must-look-at-efficiency-curves-when-choosing-your-solar-inverter/ )