No, there's no conversion for LCDs — and you wouldn't have anything approaching the resolution you'd want for a print that size anyway using a digital projector.
The usual procedure for making alternative process prints from digital files is to print a negative on transparency film. It's the go-to method for cyanotypes, as well as for platinum/palladium and so on, and the results can be indistinguishable from film. The problem here is that you want to go big — really big — and that isn't going to be cheap.
Methods that print out during exposure need very dense negatives. For that reason, you need a transparency film that can hold rather a lot of ink and produce a good Dmax. The film of choice these days seems to be Pictorico, which is available in 24" x 66' rolls. (Do I need to mention that you'll also need access to a printer that has a carriage width of at least 24", like the Epson 7X00 series?)
Black ink alone will not provide enough density; you'll need a secondary UV-blocking ink laid down as well (which means mucking around a bit in your printer settings — or getting your negatives' printer to muck around a bit in their settings). With Epsons, yellow is a good secondary. You'll also need a light stochastic noise pattern added to your image's negative in order to get enough dither for contact printing. As for the printing itself, you can build a UV source, borrow some time at a shop that still does real blueprints or screen printing, or, if all else fails, use the sun. (These exposures are long, and you can inspect them under room light safely to see if they're cooked yet. Unlike platinum/palladium, the cyanotype prints themselves aren't so expensive that you can't afford the learning curve. And the weather people will give you the UV index, which will help gauge the starting time.)
There is a much more in-depth article on creating the negative by Carl Weese at TheOnlinePhotographer.com. The negative you will need for a cyanotype is very similar to the one you'd need for Pt/Pd, and you'll need to do just as much fiddling around to find the right density and curves. Remember that you can climb the ol' learning curve using standard-issue letter-sized OHP transparencies; there's no need to spring for a roll of specialty material until you're ready to rock. Just make sure that you're using the same brand and type as the roll. (It's not something you can pick up at your local business supply depot, but it's not hard to find online either.)
Do note that you would really need to make a full-sized negative for contact printing no matter how you choose to go about this (unless you can use a modulated scanning high-power UV laser). Using a projector with anything like a reasonable-sized (read "cheap") negative would mean hours rather than minutes of exposure time, and you won't be able to tell whether or not the image is in focus until it is partially printed out.