A few observations,
I would make sure your acrylic mating faces are abraded to form a key if using a wood glue.
PVA will creep a small amount as the wood and acrylic move at different rates of expansion/moisture level, not much but enough to feel the joint after a while on something like a bowl.
Cascamite sets hard and does not creep, but I would not expect it to hold well against the noporous acrylic, I suspect any non deliberate colouring on Oak is down to iron contamination between the mixing water and/or tools used and the oak tannins.
As the acrylic is non porous I would sertainly give CA glue a trial.
What ever you do make sure that the Oak is as dry as you can possibly get it (below 8%) and that you rough cut it into suitable stock and give it time to release any stresses before you finish to size prior to cutting segments, Oak is a strong wood and will do its own thing, 1/2deg. change in all segment angles after it's glued up can break the bonds, especially as you are bonding it to a rigid substance.