Lake Braies & Tre Cime di Lavaredo same day?

Google doesn't show a route because Tre Cime are not connected by a road, but can be reached only walking. Is a nice day long hike and usually starts from Auronzo and you turn around the peaks reaching the Locatelli lodge ("rifugio", in Italian).
If you want to go only by car you can check if you can reach the Auronzo lodge, but I don't know the access rules, being a little road.
By car from Braies you must go to Dobbiaco then south to Misurina then Auronzo.

It is an overly long trip. First you have to drive over the Gardena pass and along all Badia valley to Brunico, Monguelfo, Braies. I guess we are talking about 1 1/2 hours.
Then Braies - Villabassa - Dobbiaco - Misurina - Tre cime - say about 1 hour
Then Misurina - Cortina - Falzarego, Pordoi and Sella passes to Ortisei, say 2 hours or more.

These are only the travel times. But as both Braies and Tre cime are overrun with tourists, their access roads are toll roads and cars are allowed only till there is parking places at the top. Both accesses are quite expensive - I believe the Braies one should be reserved or you will have to leave your car at the valley parking and go up by shuttle bus - this means there is little flexibility in visiting both sights, trying to do both of them in the same day could work but could not work either.

Tre Cime is seen from the loop trail, which is not a walk in the park. The loop takes at least 4-5 hours to complete. It took me 5.5 hours.

You have asked many questions in this forum and it seems that you are trying to jam too much in your itinerary. Less is more. Less time driving is more time enjoying. Also, when planning your drive, don't trust Google Maps: mountain driving is very different from highway driving. Also, be prepared for traffic jams and road closures, which happen always.

Once the parking lot for Tre Cime is filled, which occurs by mid-morning, they hold cars at the tollgate until some cars leave. Then a long line forms on the road. Unless they can arrive at the Auronzo toll gate for the Tre Cime parking by 9 am or so, they will likely have a long wait to get into the parking area.

https://www.moonhoneytravel.com/rifugio-auronzo-toll-road/

Doing both destinations would make for a very long day. I'd recommend visiting Tre Cime di Lavaredo first and then adding a visit to Lago di Dobbiaco afterwards instead of Lago di Braies. It is not as stunning as Lago di Braies, but still incredibly beautiful and much more peaceful. The lake is located right along the road to Tre Cime so it is very easy to add a stop. There is a lovely trail that goes around it. Here's how to visit: https://throneandvine.com/lago-di-dobbiaco-toblacher-see/. Have a wonderful trip!

I have spent in the Dolomites most of my holidays when I was a child; my father choose some places as bases, initially Moena and later Villabassa near Dobbiaco, and we have covered all the area in different times driving from our bases. But you have to understand that, once you have chosen a base, the best way to spend your time is visiting sights not too distant from the place you are. You can allow yourself a long drive but for us this was more an exception than a rule.

From Ortisei, Cortina is three passes distant - glorious driving as Sella, Pordoi and Falzarego are the most beautiful drives, but scheduling Tre Cime - that are beyond Cortina - makes little sense as they are too distant from your home base. If you really want to see Tre Cime, you should better stay in Cortina or Dobbiaco.

If you are in Ortisei, it makes much more sense making a trip to Sassolungo from Sella pass - no less dramatic than Tre cime. As for lakes, Lake Fedaia and Carezza are distant enough from Ortisei (lake Carezza allows for a circular itinerary, going back to Ortisei through Bolzano). Also Pordoi - the second pass from Ortisei - has some dramatic hikes by cable car. All of them are way easier than driving all the way to Tre cime.