The 1 ampere is for the whole thing, the hand controller uses bugger all.
If you extract 7 ampere hours from a 7.2Ah SLA battery (like the Jaycar guy seems to suggest) the battery will be pretty much a write-off. That said, the mount will only draw 1A when slewing both axes, during tracking it'll use much less.
Look for an 18 or 20Ah battery, ideally a deep cycle AGM one. Those can be discharged down to 30% without affecting their lifespan, provided you recharge them properly.
Cheers
Steffen.
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