unfortunately there are less open public toilets and mostly they aren't free but there are possibilities
Not for free but it's usuable
Akvárium club at the Deák square. Here join the metro lines like M1, M2 and M3. The Danube is so close, it takes to reach it 8-10 minutes by foot. This place is in the heart of the city where are many programs like conterts or yoga events.. you might check the following programs on Akvárium's website..
there is a great building with many opportunities to sit down and take a deep breath. You can eat and choose from the menu until 3PM, but the bufe is usually open until 8PM on weekdays, this is Magnet Közösségi Ház and last but not least this place is between the Deák Ferenc square and Hősök square at the Andrássy Street.
"bathrooms" are mostly at the metro stations for charge and at the public gardens and parks and at the malls as usual..
There are some public gardens where the toilets are free of charge.
These public gardens look beautiful and cozy, you just lie down if you want because it's allowed here.