It would probably be easier to use a two layer board and accept that you have ground traces interspersed with the power feeds on that side and are going to use plenty of feed-throughs. That will generally be considerably cheaper than using fewer feed-throughs on a four layer board.
Whatever you do, inductive and resistive effects will be at least as significant as capacitive, effects so any way you use the layer you should use more rather than less interconnections.