I just wouldn't use the scouring brush, it's meant to be abrasive in order to scrub baked on food off pots and pans so it's a bit overkill in my opinion.
Here is my cleaning routine for my bearded dragon's tank, I can't imagine that it wouldn't work for you, since the temperatures in my tank will be about the same as yours if not a little bit hotter.
Step 1: I fill a plastic storage tote with hot water (So hot I can't put my hand in it) and dish soap (I always use Dawn), then take the tiles out of the tank and put them into the water to soak.
Step 2: I clean inside the tank with paper towels and disinfectant. There'll inevitably be stuff that gets in between the tiles and underneath them, so it's good to clean under there and get rid of/prevent anything from smelling bad.
Step 3: I wipe down the tiles with some paper towels. Usually by the time I'm done cleaning the tank the water will have cooled down enough so that I can put my hands in to grab the tiles, if I had to guess, I'd say it takes about 5-10 minutes. Because they've been soaking in the hot water, usually all I have to do is wipe the tiles with a paper towel and any residue will come right off. If for some reason you need to have something that is stronger than a paper towel, or just don't want to kill so many trees, you could have a normal sponge set aside for cleaning the tiles.
Step 4: At this point, all I'll do is rinse the tiles off to get rid of the soap, then dry them and put them back into the tank.
Step 5: Clean the plastic tote. I have a lot of the same plastic totes since I got them on sale one time. So just in case I forget which one I used to hold the gross cleaning water, I'll take them after I use them and rinse them out with bleach water. Marking one to use each time would probably be smarter.
Optional: I'll usually wipe down the tiles with the disinfectant after I put them into the tank, just because near the end of scrubbing them the soapy water can get kind of gross, so I just want to be safe and make sure there's no bacteria that makes it through the cleaning process. This isn't necessary since I washed them with soap, but I like redundancy in these types of processes.