You can handle it in gas phase, and dilute it with argon or other inert carrier gas. (Or, when dumping into atmosphere, with nitrogen.) Such gas mixtures are used for etching silicon in semiconductor industry.
Quite a lot of unpleasant (and fun) stuff goes into making our cellphones. (I am thinking about a way for small labs to produce the stoffs “online” from less unpleasant-to-have-around precursors and purify it in a mass spectrometer before introducing into the work space; the purification step would also make it much cheaper as the ultrapure things are Expensive.)
And if you want to dilute it in liquid phase, you can use liquid hydrogen fluoride as a solvent. That dealing with liquid HF is an improvement is itself quite telling about the hell-in-a-drop nature of the material.
Also, for your collective pleasure, a handling manual for ClF3 from Rocketdyne, 1961.
http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0266121
Note the need to passivate the fuel system with gaseous fluorine. Such a friendly chemical it is.
A quote by John D. Clark, author of " Ignition! - An informal history of liquid rocket propellants" (which I heartily suggest to read cover to cover):
”It is, of course, extremely toxic, but that’s the least of the problem. It is hypergolic with every known fuel, and so rapidly hypergolic that no ignition delay has ever been measured. It is also hypergolic with such things as cloth, wood, and test engineers, not to mention asbestos, sand, and water-with which it reacts explosively. It can be kept in some of the ordinary structural metals-steel, copper, aluminium, etc.-because of the formation of a thin film of insoluble metal fluoride which protects the bulk of the metal, just as the invisible coat of oxide on aluminium keeps it from burning up in the atmosphere. If, however, this coat is melted or scrubbed off, and has no chance to reform, the operator is confronted with the problem of coping with a metal-fluorine fire. For dealing with this situation, I have always recommended a good pair of running shoes.”
On a side note, the no ignition delay is quite a safety feature. The last thing you want with a rocket engine is a delayed ignition, when you have the chamber full of fuel-oxidizer and it all starts reacting at once, instead of orderly as-introduced fashion. That thing is called a “hard start” and can lead to… well, to use a term from borrowed from Li-ion battery industry, “rapid spontaneous disassembly”.