I’m a homemaker and I work PT from home as well. I understand it’s a stretch, and it can get tiring, and tantrums feel like the last straw that breaks the camel’s back. But it’s important to put this part of the household in order first. Once this falls into place, everything else will go more smoothly.
A root cause of children throwing tantrums is sibling jealousy, and for this, I recommend a book, Siblings without Rivalry. One particular tip I liked from that book was the idea of Fairness. Fairness is not always giving equal amounts, and this is not always possible. Instead, give according to needs, and explain so.
Another is to give attention to the party that has been "hurt", instead of the one who you think is in the wrong. So whenever my younger one cries over a scuffle with gor gor, I would immediately pick him up and ask him if he’s ok first before I ask what happened (half the time the younger one is at fault). The upside of this is that now when my elder one cries (punished by dad or a fall etc), my younger son would immediately go over and ask him, "Okay?", and give him a hug.
It’s worth spending time cultivating a good sibling relationship as this has lifelong impact. And once the relationship is good, you don’t have to spend so much time looking over your shoulders. At the same time, have realistic expectations. All siblings will have squabbles, and learning how to resolve them is part of growing up and EQ training at home. Plus the younger one is usually tougher than you think.
I’ve been inspired by experienced mums to learn the art of calm discipline – listen to both sides, state the rules calmly, if punishment is required, carry it out impartially WITHOUT SHOUTING OR GLARING, after that, require the offending party to apologise to the party that has been hurt. For some people, saying sorry is difficult. If so, suggest other ways that are easier – give a hug, offer a toy, share a biscuit etc. If my 20 month old is the one at fault, I will also state firmly that he’s wrong and he has to apologise (via sign language when younger) and kiss/stroke gor gor or I give him a biscuit to give to gor gor.
Fit in one-to-one time with him, an activity that he can always look forward to doing with you at a fixed time EVERYDAY. Find out his "love language" and the things he misses most since the arrival of your younger child – obviously his life has been impacted upon. To share with you, one day I was breastfeeding my younger son in a nursing room, my elder one came in and started stroking didi’s fingers gently. Then he said softly, "I wish I’m a baby again so I can get all that love." I realised that I’ve stopped carrying him ever since no.2 came along. After that day, I would ask him, "Have I hug you yet today?" and then give him a bear hug. I also make time to play board games or do art and craft with him everyday, and I read to him every night.
Another consideration is looking through BOTH your days to see if you’re both getting enough rest and downtime. A tired person will always be more easily upset, whether adult or child. Don’t overschedule the family. If his classes involves too much travelling, consider cutting back. My elder one can be quite intense and needs a lot of freeplay and downtime, though he doesn’t necessarily sleep a lot.