are you sure you dont mean polypropylene? I distill my DCM which comes in a HDPE bottle, though even then theres no residue, but bottle caps are more likely to be PP i think, since PE cant bend an unlimited amount of times before it just breaks while PP can, within a prescribed limit.
DCM is "incompatible" with almost anything with a carbon in it because while it might not depolymerize/solvate everything, it DOES soak into virtually everything. you can go up to a big 5L bottle of the stuff and always be able to smell some through the walls, and leeching out plasticizers is why i still distill my stuff before putting into metal cannisters. but again visibly nothing.
I would probably avoid LDPE, and only stick with HDPE or PP for this, frankly id be more concerned about lye eating the polyethylene than DCM.
Oh and one last note: assuming that DCM WAS leeching plasticizers out of HDPE, it would be negligible, and literally down to nothing in the presence of water, nor would it be an issue given the limited contact it makes with the lid in the short time span you process it, which is just while shaking, and even then polyethylene is, not a spooky chem like styrenes or acrylic stuff, so even if it were entering the equation, it is benign af, neither ethylene nor polymerized ethylene are appreciably toxic, plus you already consume way more ethylene from a banana, which is like 50% of what makes them smell sweet when ripe, than you could ever hope to leech from plastic.