To use two batteries with different capacities in parallel, both should be fully charged separately before they're connected together. After that, they balance each other out both during discharge and charge.
That said; connecting a flooded lead acid battery with a AGM battery are not recommended, since they usually require different charging voltages. (14.4V@25C for FLA vs. 14.1V@25C for AGM)
Check the rated output voltage from your 110V charger that came with the powerpack. If it states 14.4V, then it's ok to parallel it with a FLA battery. If you don't care if the powerpack battery are alive after 5 months, then it doesn't matter much..
You didn't mentioned the deep cycle battery's capacity, but can you manage with only that battery? What you really should do, is to find out how much power you need each day, and how often you wish to run the gen-set, and then make sure you have at least twice the battery capacity. No matter the battery type, max charge rate should be 1/10 of rated capacity (for your powerpack, that would be a 2.8A charger), and the charging time would be 5-6hrs. (Based on the rule that one should never discharge a lead acid battery below 50%.)
What charger to use with your gen-set? I've no preference, but any charger that can give 1/10 of your total battery capacity.
I live south in Norway, and the snow usually doesn't get here before end of december...
Currently it's 22C outside and sunny, and my 1060Ah battery bank are almost fully charged from my solar array...
http://mwlmf.net/monitor/monitor.py