Is it possible to sell the apartment and take over the home loan?

If you like it that way, I would do it. I don’t think the property transfer tax should stop you. We have a similar apartment with a 100 sqm roof terrace and still prefer a house. Especially with kids, I find it cozier to be able to park bicycles, tricycles, etc.
Income fits, the apartment can be rented out for very good money.

Phew... there are still easily 360,000 EUR debts on the apartment. On top of that, now another good 600,000 for the house (purchase incidental costs alone without realtor approx. 7%) to be taken up, at 3% interest. And that with an income which is without question good, but not abundant...
There are also other costs coming your way. KNX you will certainly want again, kitchen, interior finishing.

One can also fail a legal clerkship, the bank has to assume that at first.
And here the clerkship will only start in half a year, so it is currently not running at all.

Even though I can understand the desire for a house... I would wait. The timing for financing is as bad as it hasn’t been in 25 years (interest rates high, prices high)... since you already own property and therefore don’t throw money out of the window for rent, you should wait.

Where is the apartment located, how much could one rent it for? In the Stuttgart area, one could ask for about €2000 cold rent, the installment for it is €1100. Part of it can be invested in the house. I think, with a 2% repayment and total costs of 750k, I currently expect an installment of €3500.
That would then be a burden of €2600. The first years would be tough, but if you can manage it, it would be a very good investment in times of mega inflation. Such an apartment with its own garden share is really very rare.

750,000 EUR new loan, plus 350,000 EUR old loan. That makes 1.1 million EUR debt, with 200,000 EUR equity, so 15%. It would probably be an interest rate of 4% if the bank even approves it. Is that supposed to be worthwhile? Inflation won't stay above 5% for the next 10 years, will it? And whether wages will rise accordingly is still uncertain... PS: If the apartment was only bought 2 years ago, it is unlikely to generate a return of almost 50% now (2,000 cold rent versus 1,100 EUR installment).

I assume that the bank would only play along in the current situation if you sell now (move out in about a year or so) and invest all the money directly into the new property. Otherwise, I see it similarly, that the loan amount is significantly too high compared to the income.

It looks different, meaning better, if the trainee lawyer has not only started but your wife is eventually granted civil servant status.

We see it the same way now. If we consider the reference salary with full-time + part-time work and thus reach 6k net, we can reconsider. Then you could also upgrade the full-time contract to 40 hours and would reach 6.5k + bonuses. Then everything looks a bit different.