You have thought this out very well. I have just recently "more or less officially" ended my own project on this subject.
1) The choice of 36 VDC UPS and inverters is too limited. The popular (low-cost) brands seem to be 12/24/48.
2) The DC connectors on the battery packs are made by AMP and they have a name and a color to indicate their capacity. I couldn't locate a source that would discuss sale of small quantities or samples.
As so often happens the project went a different direction. I was able to get "free for removal" some 20 year old lead-calcium cells (telephone back-up) that I had originally installed in 1986! Although maintenance has been spotty, they cleaned up well and load-tested (c/20 rate) at 100% of label rating- They are 150 amps in a 24 cell/48 volt string. The rated life-span is 25 to 30 years, and I am already old, so...
The next item to turn up was a commercial grade APC 1500watt/2200VA sine-wave UPS. I got this by horse-trading, but a new one is about $750. It has 48 volt internal pack and the plugs for adding more boxes 'o batteries. I had to hard-wire the batteries (yes there is an added fuse) because of that darn AMP connector. The internal and add-on battery packs appear to be 17 amps at 48 VDC. I programmed the UPS to believe that it has 9 extra packs attached- this is max, and the unit is too smart for it's own good- I am not enough of a logic controller guy to change anything else. Documentation is spotty- just how the manufacturer wants it, I'm sure.
In any case, the test consisted of running the window A/C in my shop last summer for 12 hours straight- remaining battery capacity was above 40%.
Rate of recharge: in this unit, the charger is efficient but not fast- takes 48 hours to bring back my (total) 167 amp pack. But float voltage is tightly controlled- no issues with water use in almost a year. I recently finished the project by running a feeder to my home office from the shop- it runs everything including lights and laser printer. I got rid of the small UPS systems that were in the office.
A small bank of manual transfer switches allows me to apply UPS power to sump pump, oil boiler, well water pump as needed. Plugging in an extension cord would allow addition of the freezer or refrigerator during an extended outage.
So, success! What remains?
1) the UPS raises constant insufferable AM radio interferance that I have yet to filter- input and output wiring carries it. Who cares? My other hobby is old radio;-(
2) maybe... I could generate DC from an old ElecTrak motor belted to my gas tractor and feed it into the battery bank, let the UPS clean it up.... and there is still that 225 amp at 36 VDC tractor pack just sitting there...