Rain water coming in through the inside of the mast?
First of all, when you say ''floor joists'' I assume that you mean the transverse frames that are typically glassed to the hull. While these may coincidentally hold up the deck their real purpose is to distribut keel and rig loads into the hull. Plastic tends to bond poorly and be more flexible so that the choice of using plastic for floor frames is a pretty poor one and one that can cause fatigue due to increased flexure where the hull turns down into the bilge. I would suggest that you use the plastic frames as mold and throughly glass (mat, then roving set in epoxy)completely over them in order to properly create a set of transverse frames.
As to rain water getting into the bilge. This is something that I too am wrestling with. I really hate keel stepped masts and that is one of the primary reasons. My current thinking is that if I ever go offshore I will have my mast cut off at the deck line and have a rigid base plate made at the deck line, close the deck partners and reinforce the deck at the mast base, and then have rigid top plates added to the top of the mast stub in the cabin. When complete the mast baseplate would be through bolted through the deck into the stub mast. The result a stronger mast (rigid end connection vs semi rigid), a mast that won''t leak into the bilge, and a mast that will stay aboard just like a keel stepped mast but a mast that can be jetissoned at sea if necesary.