Well, the MG chemicals MSDS for Ferric Chloride suggests that this would be okay.
Ferric Chloride doesn't have a flash point, so there shouldn't be a flammability risk.
Ferric Chloride will produce Hydrogen gas if exposed to metallic parts for extended periods of time, but since you are rinsing the pump after each use, I don't see a problem here.
I've never known Ferric Chloride to react with plastics, so you should be okay there.
The only potential safety issues would be if the pump fails and the liquid shorts the electricity that is flowing through the pump motor, but this risk is equal with any type of non-flammable fluid that you run through the pump. You can probably add a fuse to the power supply, just in case something like this occurs (Which is what the manufacturer should have setup for the original application in a vehicle).
Finally, hopefully you won't be running any other types of fluids (less water) through the pump. Specifically alkalis, oxidizers etc, which are listed on the MSDS as reactants to Ferric Chloride.
**broken link removed**
If you're still concerned about using a washer pump, you can check out a hobby store for their fuel pumps which are cranked manually. (The fuel pumps are used to pump fuel into/ out of the RC model fuel tanks.) This would eliminate the threat of electricity faults. I'm not sure what the internal structure of one of these pumps is (regarding metallic components etc). Nevertheless, it is a suitable alternative.