[Al Skierkiewicz]

Foto,
As to your original question on the UPS. A 2.2AH battery is a pretty small UPS. However the electronics in the UPS are designed around using that battery. Some of the things the UPS is ckecking and/or adjusting are battery health, charging current, load current, etc. By using a higher amp hour battery may affect any of these conditions besides the problem with enclosing a larger battery in the same UPS case. This is not maintenance I recommend a newbie perform for the obvious reason that a UPS generates lethal, repeat LETHAL voltages as part of it's normal operation.
As to the second part re:cold cranking amps. This is a marketing (mostly) term that relates to how many amps can be delivered by a new, fully charged battery at a specified temperature. It is more of an indication of the internal resistance of the battery at max rated current at rated temperature.
Many manufacturers include the Amp Hour rating in the model number of the battery, i.e. a MV1260G battery would generally be a 12 volt 60 amp hour battery in a manufacturers "G" case style with "MV" terminals. Batteries for new cars are chosen by size available for mounting and how much current is needed for accessories. A van with a big lighting package, power everything and front and back air conditioners will need a higher capacity battery than a sub compact with manual windows and no air.