I would also be giving up a tracker mortgage which is worth something I think! Also I may not actually need the money in a few years, so I could rent this out long term.
While the tracker is a consideration, in relative terms I don't see it as critical in your situation. Not saying I would give it up lightly, I would certainly look to see if the bank would allow you to take the tracker with you to a new property. You're saying that a new house is 850k so the tracker max would only be 150 of that.
In overall terms you seem pretty well set up financially at the moment, so I would certainly advise significant caution in not overburdening yourself with debt for the next 20-25 years. You are in a house, low mortgage, two kids, good jobs, etc. You mention the shares in the private company, any way to realise part of that to assist with a house purchase perhaps or would you want to even do that? Is it a family business, how easy would it be to realise some of that money?
Guessing your income is stable. What about your partners income? What does he do, is there potential for significant increase in his take home for example? You would need this in my view to buy at 850 based on your circumstances, and to fund the required mortgage. Might you want to reduce your hours as the kids get a bit older and other family considerations become important?
Concern I would have for you is that you seem very far short of the price of the new house that you want. Seems to me you have a borrowing capacity or around 300k give or take, may get a little more based on incomes and size of deposit were you to sell. That would leave you with 650 perhaps even 700. But you're still short by 150. You are currently saving 10k (800 per month), even if you saved a little more you won't get anywhere near bridging the gap. There is also the risk that even small house price increases will run ahead of your savings ability. Example - a 1% increase in the 850k house more or less equals what you are saving at the moment.
As an FYI a mortgage of 500k at 3.5% would cost 2,750 per month over 25 years (3,200 for 20 years). You're at retirement then, is that what you want?
I really think you need to reflect on your situation carefully, what you want in a house, family, life, etc. Appreciate you want the bigger more desirable house, do you really want the demands and sacrifices that go with it? It may tie you up for 20-25 years financially, reducing flexibility to scale back hours to spend with kids as they grow up, etc? Is there a cheaper option for another house that will meet your needs and is compatible with your lifestyle and finances?