Ciggy plugs and receptacles are horrible electrical connectors.
I'd not trust one for over 60 watts continuous and 120 watts for under 4 minutes. Those that list 120 watts or more, should have a giant asterix next to them saying temporarily capable of that.
Even if the receptacle is wired with 10AWG, which none are from the factory, the plug itself is likely 16awg and the current likely passes through a steel spring within that holds the inline glass fuse, and steel is a horrible conductor of electricity
I've experimented a lot with them, after having given up on them. for any task over 25 watts continuous.
Some designs do better than others, but they age quickly when asked to pass 120 watts continuous.
A 12v tire inflator can certainly inflate an air mattress, but they are basically high PSI, low volume pumps, an air mattress pump is basically the exact opposite.
An Airmattress with its own built in 115vac pump will likely work fine on a cheap MSW inverter, especially if it is attached right to battery posts.
While a 150 watt inverter will likely be more than enough to power it.
The '300 watt' surge rating, might not be enough, or not of enough duration to get the pump moving if restarting it when there is already pressure in the mattress. The surge rating of inverters are often measured in microseconds, and nearly meaningless unless the surge is of equal or less duration.
Lots of reports of some using an inverter to power a f 115vac dorm fridge, and they need to get a 1000+ watt inverter to handle the surge, even though after the compressor is upto speed, it only draws 62 watts.
While a MSW inverter will likely not be best for the electric motor of the built in mattress pump, the mattress is likely to develop holes before the motor burns out from having been run too long on a MSW inverter.
I've woken up on the floor more than once on an air mattress, even the good ones which do not cripple my back.
Many inverters will come with both alligator clamps and a ciggy plug, and the ciggy plug will say 120 watts maximum on it, and that is generous.
Often people see an inverter as the solution to all household appliance power needs.
Some will put huge inverters on tiny batteries. Its similar to filling a dragster with just enough fuel, to reach the starting line.
An inverter can easily reveal a weak battery. It can behoove one to idle their engine when powering an inverter, if their remaining battery capacity is unknown.
An inverter is usually only about 85% efficient.
An inverter will have a standby draw. Turned on and powering nothing, it can draw 6 to 8 watts.
My 800 watt inverter, when I turn it on, powering nothing, spikes to 0.8 amps, then drops to 0.15 amps. When I use it to power a load, then turn off that load, then it draws 0.68 amps forever after, until turned off.
So if the airmattress has a small leak, and one needs to reinflate it every few hours, leaving the inverter on just to do so can deplete a weak battery to the point it might not be able to start the vehicle.
Expect to see a spark when hooking an inverter's clamps to a battery's posts.
It can be a large spark.
Keep in mind this spark is why they say that when jumpstarting, to hook the ground up second, not to the battery negative terminal itself but nearby, so any oxygen/hydrogen possibly near the battery top, does not have a chance to ignite.
While it's unlikely to happen, it can.
The inverter spark, when attaching directly to a battery, is always a bit alarming, even when expected.
