In your example the converter-battery would be sharing the load until the converter was maxed out - what proportions each donates I haven't a clue, but in theory the AC source should load up first.
You need to forget the Honda has a DC output unless you've left car headlights on and need an emergency battery jump charge when you're miles away from a second car or something. I have the eu3000 service manual and the circuits are seperate so both can be in same circuit and used continuosly but its not worth it, the inverter side is more effiecient and why risk wearing out / accidently damaging the 12V side.
Generator AC-to-DC charging of batteries is usually absorption charge only, the last 10-15% of a full charge requires long runs times with small returns. If you need 100% charged state you'd plan ahead to let the wind or solar power handle that, the generator is used best to keep the batteries from being damaged by too-deep discharge.
A simple converter will have a fixed 13.2-13.7 voltage output - the newer ones have three-stage charging built in and your Lifelines specs say they will enjoy the 3-stage...BUT "The
GPL-8DL is designed for charging amperages up to 255 amps or 100% of the rated Amp Hour Capacity". The batteries need that high surge to circulate electrolyte, there is excess sealed in to help it squeak through the warranty period and it needs to be churned. Charging those behemoth 8D's seperately is important - Your Honda is rated for 1600 watts, so figuring 80% efficiency you are limited to an 80A converter with wire losses. (80A x 14.6 = 1168w x 120% converter efficiency = 1402w x 105% include wire losses = 1472w load)
The gotcha on AC powered stuff is efficiency - most grid powered chargers could care less how much AC they consume, maybe one of the new 'smart chargers' could be higher numbers but also probably inject a lot of noise & interferance from high frequency pulses so using a converter with its regulated & filtered output would stop that.
