As far as I can tell, the rules as written do not allow you to add multiple types to a creature, change its type to another, or change the standard listing of types. The standard types mechanic does not fit with my campaign world and I would like to change it.
DM > the rules.
This is a truth every DM needs to embrace. So embrace it. Change whatever you will. and as mentioned you'll want to consider how this'll effect certain spells/items/maybe a few class features.
But first, here, read this. It's from the Moldvay edition of the D&D Basic set from '81. It's as good as advice today as it was back then.
"While the material in this booklet is referred to as rules, that is not really correct. Anything in this booklet (and other D&D booklets) should be thought of as changeable - anything, that is, that the Dungeon Master or referee thinks should be changed. This is not to say that everything in this booklet should be discarded! All of this material has been carefully thought out and playtested. However, if, after playing the rules as written for a while, you or your referee (the
Dungeon Master) think something should be changed, first think about how the changes will affect the game, and then go ahead. The purpose of these "rules" is to provide
guidelines that enable you to play and have fun, so don't feel absolutely bound by them.
Sadly the current writers have neglected to spell this out as plainly in the current edition. It's in there, but it's all mixed throughout, implied, & it's easy to fall into the mentality trap of Rules-As-Written.