the Ah rating of a battery is a bit of a misnomer. a 1Ah battery cannot supply 1 amp for an hour. It will probably explode if you tried. It will probably supply a third of an amp for 3 hours however and not suffer significant damage. In general, batteries cannot supply more than about a third of thier Ah rating at any given moment due to internal resistance losses. Some can supply more (many lead acid and lead calcium chemistries can for instance but thier lifetime will be shortened by it) and some less. In fact some chemistries (lithiums are prone to this for instance) can explode if discharged too fast just like they can explode if charged too fast and for similar reasons. internal resistance causes heating, heating causes electrolyte to boil generating high pressure gas in the battery vessel causing it to burst or at the very least vent which causes the loss of electrolyte and the reduction of capacity of the cell. it is usually better to err on the side of caution and choose a larger battery whenever possible for best battery life.
find out how much instantaneous current the particular cell can deliver repeatedly without cell damage from the manufacturer and choose an Ah rating that can suplly the required current.