Oh Alice, it is that lovely mixture of her age and your pg hormones/tiredness making it feel hopeless at the moment. Toddlers are testing enough at the best of times.
DS1 was Cbeebied up the eyeballs when I was pg with DS2. I used to make sure I did one thing with him in the morning and one thing in the afternoon. Even just reading a book, counted as my one thing. I was shattered and weepy most of the time. Please don't feel down, it isn't you, it isn't DD, it is just her age.
Have you a secure garden she could run around in? Are there any preschools/nurseries/childminders who could have her for a few mornings a week?
FWIW I vividly remember putting DS2 (aged 2 ish) over my shoulder and carrying him out of CHester Zoo, he was screaming his head off and it was packed. I didn't care, he had pushed his luck one too many times and as he was my 2nd felt no shame whatsoever. It gets easier.
Oh and whilst pg with DS2, DS1 had an almighty tantrum in the shark tunnel of the Sealife centre and nearly got trampled on by hordes, he was lying face down in true tantrum pose. I was in tears then as well.
I have so many more stores like that! YOu are doing brilliantly, don't doubt yourself, just try and get some help/a break if you can.
Mitts, am genuinely shocked you never wanted DC, you seem such a maternal person