A points-based program that permits you to directly book the unit size you need could potentially offer some of the flexibility you want (ex., Wyndham, WorldMark, HGVC, Bluegreen). However, the annual MF for the amount of points you'd need for 2 weeks in 3BR or 4BR units per year would FAR exceed your budget. As others have mentioned, that's your problem point.
Years ago, our homeschooling family of 5 enjoyed off-season and shoulder season travel - better prices, lower crowds. Our kids often brought friends so we usually got 2BR units that sleep 8. The 2 girls slept on the LR sleeper sofa and the 4 boys slept on 2 double beds in the 2nd BR. They could choose to bring sleeping bags and/or inflatable mats for the floor if they didn't want to share. We also sometimes saved point$ by only booking 5-6 nights at drive-to locations, since Friday & Saturday nights typically require ~2x the points of weeknights.
If staying in 2BR units could work for your family, look into the locations that interest you within these systems to see if they offer 2BRs for 8 people, or 3BR units. Then determine how many points you'd need in a typical year and whether you could afford them. You'd still need to be able to plan far ahead, like 10+ months out, if you want 3BRs, even in shoulder seasons. But a good points system with resorts where you want to go and the ability to reserve directly without paying additional exchange fees would be the place to start, IMO. It would be for domestic travel, since there's not much out there in the way of a points system that includes both U.S. and European locations.