heres the risk: for HDPE as a storage vessel; virtually zero, without water, no hydrolysis occurs, and a huge mass of lye will generally disperse any moisture it rips from the air, so the wettest past is usually also the driest and vice versa. free flowing lye isnt going to be a problem.
naptha is probably not going to solvate the plasticizer in HDPE all that well, it wont soak into the HDPE and it wont solvate it well. but long term it probably shouldnt be used for storage. PP sure, but not HDPE. consult compatibility charts for the hydrate and anhydrous forms compatibility, as well as solutions, should give an idea.
Now, where things go hard is combining the two. neither naptha or strong base will cause any sort of, elimination, you could call it, naptha doesnt physically change the plastic enough that its physical matrix will release all the nasties, and lye (in solution) by itself plainly wont penetrate lower molecular weight polymers and the plasticizers, which you can imagine are oil-like compounds, visa vi the usual oil and water behaviour.
low MW compounds dissolve better than longer ones, if we are talking polymers. lye will basically let naptha leach deeper by cracking the plastic, which in turn lets lye physically erode away the plastic, so you get microplastics, probably some plastic salts from individual ethylenes cracking and combining. I dont need to go on, wether you are getting the micro plastics or dissolved plastics or just 100% now of all its plasticizer, no good. this is all avoided with no real effort simply by not conducting the reaction in a plastic container.
go to a brew shop, buy a bigass erlenmeyer, its like, the goto standard vessel for growing or mixing yeast in homebrew, i bought a 3L one from there once, any chemistry place would have you sign a declaration and give ID haha. erlenmeyers are pretty benign glassware even super big, so dont worry about raising flags looking for them. anyway, its pretty much the perfect vessel to use.