You can certainly try, but rats are incredible swimmers capable of treading water for three solid days and holding their breath for three minutes at a time. So you can flush it and it may go back down into the drain but more likely it will resist the water pressure and simply stay in the toilet bowl.
Now generally trying to flush the rat is no one’s first thought. When people encounter a rat in their toilet they have visions of the rat biting them when they are trying to defecate. This is horrifying to a lot of people and the results are quite extreme. People have attempted to kill rats in their toilets with electrocutions, flame throwers (yes real flame throwers), poisons, pouring bleach into the bowl, or spraying the rat with dangerous chemicals. Rats have been shot with shotguns, stabbed, choked, and bludgeoned with every kind of weapon that can be imagined. Rats often don’t get harmed in these attempts, they are small and agile and can generally run faster than a human. The person who gets harmed in their attempts is almost always the person trying to kill the rat. From lighting oneself on fire to shooting yourself in the leg and accidentally spilling dangerous chemicals on yourself. Simply said, if you see a rat the last thing you should do it try to catch or kill it with a weapon and definitely not with a gun as rats have high water content and will bleed out like a stuck pig all over your floor, and if it was a bullet that killed the rat there will now be a hole in your floor that the rat blood and guts will be pouring into. So leave the gun in the safe and go to the hardware store to buy some snap traps and then get a rat stopper for your pipes so the event won’t happen again. And do not try putting a mesh in the toilet as it will obstruct flushing.
If you have a rat infestation, from the toilet or not then you can try dealing with it on your own with domestic products or you can call a trained and experienced rodent technician who can come to your property and treat both the external with tamper-proof bait stations filled with commercial grade rodenticide which will cause the rat to die of dehydration and set up snap traps inside to prevent the rats from dying in the walls. That can be very problematic. The technician can then go outside and perform an inspection of the lower part of your home looking for entryways which he can then seal.
