Get your health right first. Your side project will always be there, the time you spend with your daughter will not.
Also, having an established competitor is great as it allows you to research what customers say about their experience. An MVP with a slightly different angle can become obvious from that.
No knowing the first thing about your product or market, I'll still go out on a limb and propose that you're most likely wrong. If I had a nickel for every time I thought something was absolutely key, and it turned out to be a wrong assumption - well, I could probably buy a Starbucks latte.
Hey, by all means, work hard, push forward and do awesome stuff and win, I'm all for that - but don't lie to yourself, especially not if it's impacting your health and your family.
Sure, by the time they've decided to use a new thing, they are unlikely to change again soon, and if, at the time some team is evaluating a new system yours isn't on the market, you wont be considered.
So take a deep breath, stop sweating "first to market" startup tropes, build a good product and start thinking about how you're going to sell into this market instead that's probably a much bigger challenge than building the product in the first place.
There's no first mover advantage here; there's only a "most able to convince people to switch from 90's tech to 00's tech"-advantage.
FWIW We sold video interviewing software and decided to abandon recruitment because the market had low requirements for software quality / innovativeness and high requirements for stability and track records; which was the polar opposite of what we offered.
Additionally; ATS integration was paramount for all large players. ATS providers knew that and charged for it at levels only heavily funded companies could afford.
How many real-world examples of first-mover advantage can you find?
I think that the advantage might not be as great as you'd think.