If you get glazed ceramic tile you don't need to worry about sealing the tile. You can try to find a non toxic grout and grout sealer, maybe one for use in kitchens? Then just have to hope the chin doesn't try to eat it. Caulking, the chin would definitely eat since it's soft so that is not an option. Grouting and sealing takes about a week to set up (depending on what one you use), a couple days for the grout to fully cure (depending on how humid it is where you are doing the work, could be longer) and then a few days for the sealer to dry. Grout also needs to be resealed regularly, I think at least once a year (maybe more since I don't know if pee would eat away at the sealer), during that time the chin can't be anywhere near the cage, so you will have to find somewhere else for the chin to be caged while that is being done. If you can do that then you can use wood as the base, probably glue it to the wood so the tile doesn't more and crack, causing bits that the chin could eat. One thing to keep in mind though is if the tiles ever chip or crack, which is not uncommon since chins like to throw things around, they need to be replaces as soon as possible or they can cut the chin's feet or pee can soak into the tile. If they are all grouted together the whole floor will probably need to be replaced and you will have to do the whole process over. With tiles just sitting on the floor you can easily remove any broken or chipped tiles and replace just them as needed. I mean it's up to you, if you want the added expense and work of doing the grouting and upkeep.
OK, once again tile does not replace a cool room. Having the whole cage floor tiled is not going to really do any more good then just having some tiles for the chin to sit on, only tiling the cage and grouting it is a lot more expensive and a lot more work. A much better idea would be to have tiles you can remove and store at least a couple in the fridge so you can switch tiles out for cold ones if the AC is out do to a power outage. If you are heading out and it's a warm day you can put a few cold tiles from the fridge into the cage in case the power goes out. If you grout and seal all the tiles into one solid floor you can't take it out to cool it.
During a power outage you can put some tiles in a cooler with some ice and change them out as they warm up, depending on the temp every half hour to an hour. You can also put bottles of water in the freezer and wrap them in fleece (so the chin can't chew them) to put in the cage for added cooling. You can also make something like this
Keeping chinchillas cool for cooling if the power is out. If the power is out of more then a day when it's hot though you will likely need to find a hotel or some where that has AC to take them to. These tricks are only really for temporary cooling, like for a few hours.
One other thing I was thinking about, with a tile floor it wont be as safe for a kit as fleece. Kits climb and like to climb the cage and then jump down, a bad landing on tile will be much worse and more likely end in an expensive vet visit (or worse) then falling on to soft fleece. Most chins outgrown climbing the cage as they get to be adults though.