The arguing is modelling behaviour on how to talk and resolve conflict so I would look at what the argument is. I grew up in a shouty household, my Mum would shout at us for misbehaving, never talked to us, just shouted. Same with my parents, any disagreement was immediately raised voices, sarcastic tones etc. It took me a while to realise my friend's parents didn't do this. They were able to diffuse situations with humour and love for each other. Really opened my eyes.
If your Dh is thinking that if you moved location you would get more help, how true would this be, realistically? Are his parents retired or working? If you visit them or they visit you how much help do they give?
If your DD is not exhibiting this behaviour in school then it is because she knows the rules at school and the consequences. Do you set out expectations before you leave the house or in the home, that you are expected to behave a particular way ie sit at the table to eat lunch. What would be the consequences for not doing it?
Does she do these tantrums at other people's houses at play dates? Or is it just at home/with you, her parents?
Her behaviour will also be modelling behaviour for your 5 year old too. It does sound like you are both exhausted from working and relatively young children. Any friends who can do a play date swap with you? Any activities you can enrol them in jointly/at the same time to get a break?