No it shouldn't dissolve and as long as you are using a fine enough filter (stuffed cotton balls, coffee filter, etc) then you should not be getting much MgSO4 coming through. I never intended to drink the alcohol so I couldn't tell you for sure.
If you do this and would like to drink it, please double check its clean. Pour some of the alcohol on a clean mirror or glass dish and let it evaporate. There should not be any residue.
WARNING: Of course this is just for creating (dangerous) lightweight alcohol stove fuel and I would never recommend you come in contact with alcohol as you could explode and die.
Forgot to mention… this is 'kind of' important and hopefully obvious. Hardware store denatured alcohol has got poison in it that you do not want to drink cept maybe after fractional distillation and who knows what else. If you are going to make lightweight drinking alcohol to give to chimps then start off with spirits from the liquor store.

