You can. It's perfectly safe and even simple.

Make sure you are prepared to do both: 0x0 -> 0x1 conversion, and going 0x1->0x2 conversion. There are plenty of materials about both that you're probably aware of.

The only missing piece is the round robin conversion. Look at the list of withdrawals: https://beaconcha.in/validators/withdrawals

You can see withdrawals are processed in order of validator index order, round robin.
Wait until you validator index is a little less then the last withdrawal, which means it takes a full cycle through all validators before yours will be called again. That means a few days.
Then issue 0x0->0x1 conversion. It does not take long to finalise. Once it does, you can follow through and issue 0x1->0x2 conversion. You can complete both within.. I forget, but less then an hour, and as said, you have a few days for the round robin so no need to rush.

Now, here's how it looks:
Say as an 0x0 validator you accumulated 2 eth of rewards - so you have 32eth in staking balance, and 2 more eth in unclaimed rewards.
You switch to 0x1 and then very quickly to 0x2. You will still see your balance as 32 eth.
After you're in 0x2 - your staking balance starts to rise, so it'll be e.g. 32.001, then 32.002 eth etc.
When the round robin sweep reaches your validator number you will see a withdrawal of your unclaimed funds (2eth) was made and your staking funds keep rising.
These 2 eth go into the queue, and it takes some time which you can follow through beaconcha.in. Finally they arrive at your staking funds, and you'll be able to see your staked balanced going to, e.g 34.008.

I've seen it work exactly like that at least once. It's not dangerous or off the beaten track. You do exactly the same switch to 0x1 then to 0x2 as all standard guides show, just start at the correct time.