I normally sand to 12000 mm, finish with CA then follow the CA with plastic polish.
I have worked with about 40 Gallons of Alumilite (making blanks not turning them) and have never had a problem with it becoming soft while cutting the block into blanks or finishing or turning. Make sure you are mixing it by WEIGHT not quantity and although once set the blanks can be worked with in a very short time I normally let the blocks cure for about 3 days before I cut them up. On occasion I have had the top part of the block (the part exposed to air) stay a bit sticky but once I cut into the block it is dry and solid. Also, when I am making the Amalgam-Mutt" blanks (the resin & wood blank mix) some woods like mesquite and Banksia nuts, "weep" (as in release some sticky stuff after the block has hardened) IF and WHEN this happens I leave the block on a shelf for 6-10 days and by then the weeping solidifies. Although I agree heat may soften alumilite it would take A LOT of heat...finishing does create heat but it should not create so much heat to soften the blank (if it creates that much heat your fingers should be burning before the blank starts softening (assuming the alumilite was misxed correctly).