Thank you for your email. It sounds like you have a lot on your plate with 3 young children and your husband being away.
Your 3 year old is crying out for attention from you and senses that you view him as a "brat". I always discourage labelling children as they tend to live the labels they are given. Children are extremely perceptive and he is acting out because he knows you respond differently to him than his older brother.
You are comparing a 5 year old with a 3 year old, and there is a vast difference in what they are able to do especially with a young baby. Instead of separating your children let them both help look after the baby and gently manage and supervise them. Your 3 year old could be responsible for picking the toys, clothes the baby will wear. Let him hold the baby while sitting on the couch next to you and so on.
Your son is not deliberately trying to upset you and you can turn this around by doing a couple of things. Firstly you need to change the recording you have of your 3 year old in your head. Make a list of all his positive traits. He is cute, clever, able to run fast etc. Focus on these and tell him. Stop yourself from comparing your children and re- evaluate your expectations of your children.
Secondly you need someone to help you care for your children so you can take some time out and de-stress. Ask a friend or relative to pop in for an hour a day so you can take time out and do something you enjoy which fills your soul while they care for your children.
Then you need to arrange for your children to see their father as soon as possible. Change and uncertainty can make children feel very insecure. Your sons need their dad and once you are back with him encourage dad to be as involved as possible in their life.
Make a point of taking your 3 year old on a date where you and him do something fun together that he is able to choose. Parents should have individual dates with each of their children at least once a month. In addition spend ten minutes with each child holding them on your lap and telling them how special they are to you and how much you love them.
If you feel as though you want to 'destroy' or hit your children speak to your family doctor or a counsellor. Your children don't need punishment. The next time your 3 year old is acting out take him onto your lap and tell him you understand he is angry and frustrated and would like to help him. Maybe you could do an angry dance with him. When he knows you understand him and are sensitive to his needs, he will calm down.
Good luck. let me know how you all are doing.