Yes, you can have Infuse run on one device to index everything, then other devices would be able to read the info from iCloud. If you have a choice, the Apple TV is a good option for this since it can be left on for awhile without worrying about battery life.
It won’t hurt anything if you have another copy of Infuse open at the same time, as Infuse would simply update and/or discard any duplicate records it receives.