In my experience, and echoed by others, there can be water loss from the grout into saturated sand as well as dry soils that can make installation of a cage after grouting difficult. One solution is to redrill and regrout the pile after a short time; often it will then accept the cage. You haven't told us what the soil conditions are, but I assume you may have weak clays over sands. You may be able to get a cage in if it does not go below the bottom of the clay.
A single bar or bundled bars can be installed through the hollow stem of the auger in most any soil. Cages are nearly always installed after grouting.
If seismic forces, tension loads, swelling soils, and lateral loads are absent, a light cage about 15 feet long to handle incidental moments may be sufficient. I have seen this done on thousands of piles in the midwest. Check your local building code, of course.
Local ACIP contractors can probably tell you what is feasible after they see the boring logs.
I would be concerned about the water table only if it is above the pile tops, which can lead to bleeding and cement loss. The ability to be installed in wet soils that would cave in open holes is the main appeal of these piles.
RE: the timber piles. Excessive soil removal during installation of the ACIP piles could undermine the prevously installed timber piles. Lateral displacement during driving of the timber piles might damage the previously-installed ACIP piles. Be careful!