Inverter Generators for the most part are Pure Sine HIGH Frequency, they provide cleaner & more regulated power but also suffer the issues of normal High Frequency Inversion tech. Some of the higher end ones have Low Frequency inversion which is more capable and better for anything with a motor.
Standard generators are Modified Sine Wave, which is not that good for motors and many fine electronics can be negatively affected.
Sine Wave Types and the Differences between Hi / Lo Frequency inversion is a rather extended discussion and can often generate a lot of "noise" because there are a LOT of misunderstandings and confusion.
Any Inverter/Charger system including AIO's which can take an external AC Power Source (either Grid or Genset) will accept the power, provided it meets the required Parameters for operation.
Using 120VAC as an example, it can vary between 58-62hz frequency and typically the max boundaries are 55-65hz beyond that it's "too dirty" and equipment cannot operate properly. The Voltage also has to remain fairly tight between 109V-128V, beyond those limits, again equipment will not operate properly. If either goes out of "bounds" damage can result, most Charging Systems will disconnect when outside of permitted limits.
An Inverter/Charger may be capable of doing "Pass-Through" power from the external source to the Home" while charging batteries as well. These cannot correct incoming power variations nor take Mod Sine and convert it to Pure Sine, so more or less, what comes in is what goes out while charging is proceeding. I have never encountered any information related to the impact on the battery cells as the charging system is converting to DC.
One must be aware that whatever the Genset is putting out WILL pass through the whole system while charging the battery bank. This CAN have a deleterious effect on electronics, motors & devices and could even cause damage.
Example:
I use a Samlex EVO-4024 Inverter/Charger (Tier-1 product and NOT cheap).
I use a Genset backup that is connected to the generator L5:30 (120V/30A) outlet.
When I charge the battery bank, I am pushing 24V-28V/80A at the batteries.
That is pulling 120V/24-27A and approximately 2700W +/-200W. This floats a bit relative to loads etc THIS IS CONTINUOUS !
I am using a Champion 41552 7200/9000W "Construction Generator" for charging up and it works fine BUT this unit can tickle the edges of the allowable specs and that causes the Inverter/Charger to reset & disconnect inbound "bad power". The MOD SINE passthrough is very noticeable (fridge motor, microwave, show it blatantly) but I am extremely cautious what I use while Gen Power is on.
Large Inverter Generators are had to come by and a decent one is quite expensive. The smaller units like the Honda 2000i (really good but $$) are capable of 2000W Surge but only 1500-1650W continuous (pending on models it varies a bit). You can link Two to double output.
Brands like Hyundai, Energizer and MANY more (westinghouse) have NOTHING TO DO WITH THE BRANDING ! Energizer Battery Co has NOTHING to do with generators, same thing for Hyundai Industrial. These are Chinese Made (legally) but are "junk". BEWARE OF BRANDING !
They are Imported by Midland Power Inc.
http://www.midlandpowerinc.com/ Other "Brandings" are imported by others.
I CAN tell you that Champion Power Equipment has always been reliable, I've used their equipment since the early 90's and no issues. I am "religious" about oil changes & use only 0W30 Synthetic Oil and do proper maintenance. My "Beast" gave me the power I needed to build my home, run my compressor (3HP/2-Stage/25Gal) and my Mig Welder as well. LOTS OF HOURS and it is now 6 years old. My other gear no issues either. They do make Inverter Generators which are quite good as well, including models which can use Propane (bonus it lasts and won't sour).
Understand the difference between high frequency and low frequency inverters with this quick article.
www.magnum-dimensions.com
Hope it helps, Good Luck