Congratulations on your growing family!
No experience with tiny houses per se, but when my son was 3 my DH and I bought a ~200sq foot camper (27 feet long, 8 wide) with the thought that we might take a year and see the country before DS started school. A 2.5 week camping trip the following December -- in SC, so a climate about the same as what you're living in -- convinced me that if we did, I'd strangle our son. Obviously a lousy idea, so we scrapped the year-long travel idea and have stuck with (and very much enjoy) shorter trips close to home.
My son is high energy, but he's, you know, one of those kids in the top-10 percentile of energy, not one of the ones in the .0001 percentile -- more energetic than average, but not crazily so. However, at 3 (or 4, or 5), being confined to 200 sq ft of space 2 hours or more before his bedtime after a long and busy day (of vacation! But there's no other way with a small kid...) was more than I wanted to stand. And, sure, you can take a 3 year old out, warmly dressed, in the dark when it's 30 degrees and windy out (it was an unseasonably cold winter, our first trip. We've done about the same camping trip every year since and they've all been much easier because of warmer conditions), but it's not really fun to have to do so (and no fun at all if it's precipitating. We spent several hours one evening entertaining DS in the campground laundry room because ... well, it was what worked).
All that said, (a) your child may be lower energy than mine -- my son is my DH's 3rd kid and he (DH) still rolls his eyes at how much more energetic DS is as compared to my stepkids; (b) the first year and perhaps even the second will be easy -- from about 2-8 months of DS life, the only parts of our home that were truly liveable were our bedroom (which he shared with us), laundry room, and 2 bathrooms. The other 2 bedrooms were cram full of all our living room (etc.) furniture, and the kitchen/living area were being renovated. Indeed for several months we didn't have a working kitchen (we did have a fridge and microwave out in the "construction zone"), and a grill on the deck plus of course running water in the bathrooms). And the construction zone really was one, bare subfloors, studs, etc., and it was totally fine because, as noted, infants aren't self-propelled (not having a kitchen was a nuisance, of course, but not a big deal. Way easier than being in camper with a 3-year old, for example!).
Long story short, I'm skeptical, but not in a way that says, "Run out and sell it now!"