As a caregiver you absolutely MUST take a break. It's not optional. It's a must. Remember that nothing changes until your needs become a must!
Do NOT engage in the temper tantrum. Give it absolutely no energy. Leave the room. Walk away. Make it clear that you are not going to listen to the temper tantrum. You can say this calmly, but it has to be said.
If your loved one tells you that you don't love him/her, gently take a hand when the tantrum has calmed down and say, "I do love you. In fact I love you so much and want to be here for you, that I have to take a small break in order to be able to give you the best possible care." Leave it at that and don't get into a discussion. You do not have to justify taking a break from caregiving.
Make time for happiness and practice scheduling time for you every day. It doesn't have to be an all day event, but doing something for you each and every day will set the standard and eventually your loved one will come to be more accepting of your self care and personal boundaries. Schedule your time just like you schedule all other appointments.
Finally…understand that the first few times you actually follow through, you're going to feel guilty. You're going to feel like you've done something wrong. You haven't. This is just the chatter in your head and the only way to quiet it is to practice the art of self care. Learning ways to manage guilt is one of the keys to successful caregiving.