Just so you're aware unless they've changed it the Granite Falls fish do not count in the aspiration part where you have to collect 20 Different Types of fish (not sure if the Pufferfish would count either) the last time I tried this they needed to be the base game fish. So I would suggest completing the aspiration first and then send your Sim to Granite Falls to get the fish to complete the collection.
And no the fish don't need to stay in the inventory like any collection the game registers it in the log and remembers what you've cost. I would suggest writing down each type of fish or write out the complete list of available fish so you can mark it off. Certain fish are easier to catch with certain types of bait to check out the list above to keep track of that.