Don't be married to DCM, that reaction would work in MeCN, THF, DMF, DMSO, etc etc etc. However if you're starting with the allylamine hydrochloride then of course it won't be soluble in organics, and you'll NEVER get all the water out of that aqueous version, so it will always foul the reaction. Don't even try.
What I would do is take the powder, and add it to a solution of organic solvent and 2+ eq of a base (TEA, DIPEA, DMAP, whatever) and let it stir until it dissolves. Then add your acyl halide. This polymer amine salt COULD be soluble in DMF or DMSO btw. Keep in mind your basic salt might precipitate. You want to choose a base that is a stronger base than your allylamine.
I think two phase system is terrible for polymers. You'll get foaming, emulsions, poor conversion etc. It will be horrible. Schotten-Baumann condition is for small molecules because they are great at seperating between phases and staying there.
P.S. Even for the polyallylamine powder, consider drying the heck out of it with P2O5 setup, low heat + vacuum, or a dessicator. Its probably hygroscopic itself.