There is no way I would do this.
It's unfortunate that your wife is reacting this way. Understandably, she is disappointed that you can't afford the dream house.
You have a fantastic savings pot - seem to have made up for the foolishness of earlier years.
Your calculations leave no scope for emergencies - the boiler has broken, car broken down, mortgage rates increasing. Not to mention the unthinkable - one of you here sick and can't work.
You both, your wife seemingly more so, are in need of some reality.
You say you would borrow from family - how would you repay them? Is that included in the repayment figure about?
Whilst I understand the desire to go all in on the forever home you do not want to overstretch yourselves to be house rich and cash poor.
Money problems is one of the leading causes of issues in a marriage - which you are seeing already. Don't go in any deeper.
Perhaps, in the fullness of time, your wife and/or yourself could work towards promotions etc and get the dream home later...