So if I take my Eeveelution that is level 17 to the move deleter and make it forget all its moves but Last Resort, will it be usable in battle without using other moves first?
Last Resort will fail if it is the only move in your set, you must use all other moves in the set first, and it will fail of it is the only one. However, if you only have 2 moves you can easily use the first move then spam last resort to your hearts content.
Sources: Personal Experience
http://Pokemondb.net/pokebase/65104/can-i-use-last-resort-if-its-the-only-move-on-my-Pokemon
http://Pokemondb.net/pokebase/45924/if-last-resort-is-your-only-move