Firstly don’t worry at all about feeling judged. There isn’t a single parent who hasn’t had to deal with a tantrum. It’s normal human behaviour to look at a child who is screaming or crying but that doesn’t mean you are being judged, you can’t possibly know what the person looking is thinking, they could be thinking “gosh, I’m.so glad that’s not my child for a change”.
I agree with pushing her nap to after lunch. Take her to the
playgroups/swimming Sessions/tumble tots. Whatever you can find. Let her have a small nap in the morning but then wake her and get out. It may help you to see you’re not the only one who is tired and struggling. If you are Bfing, are you getting to any BFing Support Groups like Le Leche League?
Once you’ve been out, give her lunch, if she’s still awake and then let her nap. My DD would often fall asleep on the way home from being out in the morning so would end up having a late lunch. Do what works for you.
The No Cry Sleep Solution has sone advice on early mornings which might help, especially as the 4am is the new 5 
Definitely get her some reigns too. It’s handhold, pushchair or reigns. You have to be firm, there is no compromise or she’ll get hurt. At 18 months she won’t know the dangers of running into the road.
As for activities, have you tried cooking or baking with her? You can get child safe knives which they can use for chopping things like bananas and both of mine would be happy to stir mixtures and help with baking.
How about buying 2 types of pasta, say one twists and one tubes, mixing them in a bowl and getting her to let them out?
Could she match socks? If you put a line of single socks and give her the matching pair and ask her to match them up?
Would she put washing in the washing machine or take it back out at the end?
Have a look in the local charity shops too. Things like simp,e jigsaws or giant beads for threading are usually cheap.
There’s a few more ideas here OP.
Let us know how you get on 