Lightest weight deep cycle that will suit?

you might look at the XS series battery's, 26 pounds, www.speedwaymotors.com/XS-Power-D925-12-Volt-Deep-Cycle-AGM-Battery-65-x-693-x-492-Inch,33020.html

First question have is why use an inverter? It complicates your battery issue, with little benefit. Just use a 12v car accessory plug mounted inside the trailer for charging.

"One" normal 12V deep cycle battery connected to a battery maintainer (like a Battery Tender JR.) is going to work just fine for your limited needs. Leave the battery stored inside the trailer, and it should be stored "plugged in" to the 110v system 24/7. The battery maintainers are "smart chargers" and shut off when the battery is fully charged.

I'm late to this thread, but in case you haven't bought your battery yet...

To accurately choose the "right" battery, you need to add up all your electrical usage per trip, in amp-hours, and buy a battery twice that size (because discharging a battery below 50% drastically reduces its lifespan). That's assuming there's no charging available during the trip, but if you have a charging source, then you can subtract the amount of charging amp-hours available from your usage amp-hours and buy a smaller battery.

To figure it out: Your fan has a data plate on it that tells you how many amps or watts it uses (if it says watts, just divide by 12 to get amps, because watts = amps x volts). Let's say it takes 4 amps (48 watts) like the fan in my furnace, and you want to run it for six hours per day. That means you have to come up with 24 amp-hours per day just for the fan. If that was the only electricity you'd ever use, you could buy a battery with a capacity of 48 amp-hours for every day with no charging. The very-commonly-used 75 amp-hour battery would give you a day and a half before it got down to 50%.

To be able to go camping confident that you won't have to, say, stop using the fan because the lights are more important and you can't power both, or worse, not noticing until the lights turn brown and the fan dies, you need to make a list of all electrical appliances, their amp draws, their projected usage hours per day, add it all up and double it to come up with the right size battery for one day's no-charging use. Six hours per day on a 2 amp light bulb would require another 24 amp-hours of battery capacity per day (6h x 2a x 2 [for the 50% discharge margin]). So the fan and one light bulb would require a 72 amp-hour battery for each day of usage without charging. It adds up fast. LED lights help a lot, but motors and heating elements will suck the power up.

All the batteries you're likely to consider are made mostly of lead, they all use the same chemistry, so they're all very close to the same weight for the same capacity. There just isn't a way to save useful amounts of weight without losing capacity, unless you add charging. For the record, sealed AGMs actually give you less capacity (and much higher prices) per pound, because the way they avoid gassing is to lower the charging voltage, and that means that they are never charged to the same level as a wet battery of the same weight (and which also makes them charge slower, and makes your choice of charger critical, both to prolong the life of your battery, and to prevent overcharging them to the point where you blow the gas safety valves, which ruins the battery instantly, if not causing an explosion or fire). The most efficient power-to-weight-to-dollar ratio comes from plain ol' wet lead-acid batteries, even though they are still heavy. No one's invented a styrofoam-acid battery yet.

It seems to me the best way for you to save weight would be to add charging, probably from the vehicle, since solar panels and generators also add weight. Make sure to use an isolator switch (of the proper amperage to match your alternator), which switches your alternator over to charging the camper battery after it's done charging the car battery (any auto parts store). If you just hook a wire from the car battery to the camper battery to charge them both at the same time, one will always be undercharged, unless they are identical batteries (and car batteries make lousy camper batteries). Also, make sure the charging wire is plenty big enough. A too-small wire of that length will waste a bunch of your charging power heating the wire instead of charging the battery, due to voltage drop. Good luck!