So I have installed a few KISAE products and have found them well made, and on the occasion where a failure was partially caused by the operator error, their support was excellent, they sent out a new unit.
First lets answer your questions
KISAE Abso 2000 Compact Pure Sine Inverter/Charger Charging specs.
55 amp max input, with four battery settings, Flooded, Gel, AGM, and Fixed Voltage.
The AGM setting gives you a Bulk, of 14.3, Absorption of 14.3 and a Float of 13.4 The unit does not have the capability of temperature feedback.
The Lifeline spec calls out for Absorption of 14.3 and Float of 13.3 volts at 77°F
So the charging specs are pretty good match for your battery with the exception of one thing, no battery temperature feedback. If your charging will only be done around the temperature of 77°F, this would not pose a problem. See chart below
Now, on to the inverter part. It should be noted that while this is a pure sine inverter, it is high frequency inverter. For the explanation of the difference of a high frequency and low frequency I will got this link Inversion Methods Explained: High Frequency vs Low Frequency
from Magnum Energy.
IRON CORE TRANSFORMERS AND FIELD EFFECT TRANSISTORS
There are two distinct types of industrial grade power inverters distinguished by the size of their transformers, and the switching speed of their transistors. The ability of an inverter to absorb the electrical surges inherent in certain loads like motors, pumps, and torque-related tools is directly proportional to the physical amount of iron present in the transformer. Size and tolerances of the transistors used in the inversion process, and the speed at which they operate determines the classification of high or low frequency.
INVERSION METHODS EXPLAINED
High Frequency Inverters (HF)
The large majority of inverters available in the retail market are high frequency. They are typically less expensive, have smaller footprints, and have a lower tolerance for industrial loads. HF inverters have over twice the number of components and use multiple, smaller transformers. Their application is appropriate for a wide variety of uses like tool battery chargers, small appliances, A/V and computers, but have a decreased capacity for long term exposure to high surge loads like pumps, motors, and some high-torque tools.
Low Frequency Inverters (LF)
Our UL-listed, low frequency inverters and inverter/chargers are the pinnacle of electrical durability. The massive iron core transformer is aptly capable of absorbing surge loads because of the “Flywheel Effect” inherent in the physical amount of a transformer’s iron. LF inverters have larger and more robust Field Effect Transistors (FET’s) that can operate cooler, in part due to the slower frequency of switching required to produce AC power. These inverters are feature rich to include the optional ability to hardwire additional external GFCI outlets, input of multiple DC voltages, provide regulated dual output voltages (120/240VAC), and integrate chemistry appropriate battery chargers and transfer relays for shore power.
Doesn’t operate well with high-surge loads like pumps and high-torque tools
Runs well with high-surge loads
As you can see, the high frequency inverter would meet your needs being that a 700 watt microwave would be a small appliance, it should handle the PC, and chargers also. While the peak rating on high frequency inverters is generally higher than low frequency inverters, the time frame supported is actually shorter, so a 2000 watt would probably be a good fit.
A possible better choice would be the KISAE Abso 2000, 12V Pure Sine Inverter/Charger
This is a low frequency inverter, it will do support what you want to do with an inverter. The charging parameters are pretty much the same 14.3 Absorption, and a float of 13.45 (slightly higher) Lifeline AGM float 13.3. But more importantly it has the capability to accept a Battery Temperature Sensor option, which should better track the charge parameters.