That seahorse is going to grow to 8"-11", but not if you keep it in the 5. If you keep it in the 5 for anything longer than a quarantine/growout period, its growth can be stunted and its lifespan shortened.
Not to mention, a 2" seahorse needs to eat at least 3 times a day, and 3x a day feedings of meaty foods like frozen mysis shrimp (or live if it won't take frozen), will quickly deteriorate the water quality. You're looking at water changes on a daily, or every other day basis, even in a cycled tank, to keep nutrients down.
For reference, we recommend a 29 gallon tank MINIMUM for the large species of seahorses, and if this is a true kelloggi, it is going to need an even larger tank for adequite swimming space.