hiya Smidge!
i wasn't a SAHM, though i was home all the time - i work from home. we had a nanny for a bit (a young, energetic uni student who came in for several hours every day,) but even with her there, it was often a very long day for both DD (now 5) and myself.
so like Dude, i really set some limits, not only with the kid, but with the nanny, my husband and virtually everyone who looked after her for significant time. it was a variation on "house rules" - every adult set their own expectations and was clear about them. DD learned what she could get away with in each person's care and everyone felt free to take a break if they needed it. she really was unbelievably exhausting - i thought i was just a wimp for the longest time!
i'm personally kind of terrible at pretend play, so one of my tricks was to co-opt the kid right in with whatever i wanted to do. she did lots of kitchen prep and vinegar-based cleaning around that age! she actually did a little of my work, too - on slower projects i'd sometimes hop her onto my lap and get her opinion on advertising layouts. (which i would not always take, obviously - but that was good, too - we had great conversations about the colour wheel and page balance!)
i'm not sure if this will work for your situation, but maybe you could shift Dad-time to after work? if you handed your bean over at the end of the day for a snuggly bedtime, you might save yourself the morning aggravation? if that doesn't work, maybe ask him to switch it to something quieter, like a story time with a definite start/finish? it might be the arbitrary end of the playtime that's setting her off so badly.
for our family, 3 was a pretty physically demanding year, but it did pass. we have other crazy & dramatic stuff now, but i do feel better equipped to deal with it since i'm not quite so bone tired! best of luck!