High sap diapers swell a lot but get firm when wet. High pulp diapers start out a bit thicker, don't swell much (if at all), and aren't as firm when wet. So it sounds like you're looking for a high pulp diaper.
The problem there is low sap also means lower capacity, so you're going to have to sacrifice a little somewhere I think to get the rest of what you want.
I would suggest the older Bambino line - Classico / Bianco / Teddy. They're a medium-sap diaper that doesn't swell a lot or get very firm when wet, but is still pretty good capacity.
The current Confidry 247 are fairly close to what you're looking for also.