Does Storage Studio check my cloud accounts for duplicates while on battery?

Yes, unless the battery is low. Checking a connected cloud account only reads a list of file fingerprints over the network, which is light work, so Storage Studio lets it run on battery after a short pause in your typing. Scanning this Mac is different: it reads the whole disk, so it waits for mains power.

Both still wait if the Mac is hot or in Low Power Mode, and each cloud account is checked at most once every few hours, one at a time. See why duplicate scanning is waiting.