Can only use small sump pump basin in crawl space, what can I do to prevent short cycling?
Your &quot;basin&quot; need not be sold as a &quot;basin&quot; -  the good old fashioned, fits very easily through limited access, pile of bricks or blocks that you use to line a hole that you dig (of whatever size you like, since you assemble the wall in place) will solve your diameter problem.
The other obvious solution would be a different pump with an adjustable float, or a secondary float switch for the pump you have that would prevent it from turning on at 7&quot; when set in a deeper basin.
If you are dedicated to pre-manufactured sump basins, install several of them in close proximity connected by 3&quot; or 4&quot; pipes. The large aggregate you propose will fill much of the void space you propose to add - if you are digging that much hole, line it with bricks to get most of the storage capacity from the large hole (or use multiple connected basins.)