Dick,
I don't think it would work.
The problem is that when the Beaver brains see 120v at the plug, various circuits are powered from the inverter "pass through". Examples are the TV, some lights, outlets with computers plugged, etc. And, it takes only a momentary mental lapse to plug in a hairdryer or something. When 120 is not available, everything automatically switches to the inverter drawing from the batteries. There is no manual control over that switch. Setting the inverter at 5 amps might work, but I think the margins are too small when the power from the 1000 is only 7.5 amps. It is too easy for the TV or something to need a little more than the 2.5 amps left and draw it through the "pass through".
Several times I've managed an extended stay in someone's driveway on a long cord to a 15 amp outlet in their garage. I set the inverter to 10 amps. You have to set it before you plug in because the charger, as soon as it sees 120, will try to top off the house batteries and will use a brief spike as high as 20 or 30 amps to do it. Pop goes the 15 amp breaker.
When doing 15 amp hook ups, the household rule in this Beaver is to plug nothing more than cell chargers and laptops in. Nothing, unless, you start the Onan and let it get through it's one or two minute startup cycle. The brains in my Beaver give priority to the Onan over the plug. No mater if it is 50 amp or 15 amp hook ups, the Onan gets to do the work if it is running. So, when DW BJ wants to get dry hair or cook my dinner in the Sharp, she can poke the Gen Start button. If we forget to start the genset, the garage 15 amp circuit will quickly blow, the inverter will take over, the batteries will drain down and the auto start system will get the Onan going anyway. These Beavers are pretty smart.
You could test all this theory. My primary Honda 2000 accessory is an inline amp gauge designed to measure appliance loads like toasters. Set your charger to 5 amps, plug the Beaver into any 15 amp circuit with adapters though the inline amp gauge and watch. If the load stays below 7.5 amp, the 1000 would work.
All my theory and (hopefully) knowledge on this comes from a long discussion on rv.net a few years ago. At the time I was trying to figure out how to camp in my kid's driveway and I was considering a trip to Mexico. It is rumored that Mexican 120v power could be less than the Beaver likes and many campgrounds only provide 15 amp hookups. One of the benefits of the independent charger is that it separates the Beaver from the grid. Bad things could happen on the Mexican power grid and the worst would be the charger burns up.
Buy a charger anyway. It is amazing how fast you can make friends in a campground when someone's car battery is dead. When we camp host, I loan it out a couple times a month, usually to tenters.