Financing new construction with an existing mortgage on the property

With the preliminary draft, you can estimate how high the construction costs will be. Demolition 30k and roughly 3000 euros per m2 of living space for new construction. With a hillside or expensive area, also gladly 3500 - 4000.

Demolishing a house that is only 30 years old and then renting during the new construction does not seem sensible to me. How many square meters do you want and as an extension or an additional storey? Depending on the interest rate, it does not make sense to save up equity instead of paying off the debts.

a house only 30 years old

with not even 50 sqm for 3 people? No matter what you do, you won’t be able to live in the house either with an extension or a demolition. The renovation with an extension could indeed be more expensive than a new build. In that respect, it probably makes sense (without knowing more about the house than already known).
After all, the OP has been planning for over a year (with an architect?). That was also your advice, to go to an architect.