Bluetooth is an audio standard, not a video standard.

Likewise, you want to run cables if you want it to work. Up to you, but wireless is a joke with video.

Bluetooth operates at up to 25mbs. That's great, but uncompressed 1080p video runs at about 3Gbs. Or, over 100 times faster than Bluetooth supports.

There are wi-fi solutions, which work so-so. There are HDMI dongles which can work very well, sometimes, but are not guaranteed to work.

Then, there are actually wires. If properly installed, they are nearly 100% reliable and will work for years with all laptops and should just be plug-in and present.

Other considerations: You will need power at the projector. So, while wiring for power, why not add the cabling for video?

When considering the projector, you shouldn't make a call on lumen requirements without knowing the size of the screen first. You also want to consider throw distance in everything. Laptop location in regard to the screen will matter as well.

There are a lot of questions I would be asking before recommending a projector, but there are some decent LCD models which I would be considering.

http://www.projectorcentral.com/projectors.cfm?g=1&hide=0&st=1&r=&br=5000&br=7000&c=&w=&ar=Wide+%2816%3A9-10%29&db=&zr=&wt=<g=&ll=&mfg=&p=&wr=&dt=1.0.0&t=&pjl=0&pjw=0&pjh=0&td=&is=&i=d&tr=&tr2=&oop=1&sort=pop&sz=15

You can see that the price ranges widely from under $2,000 to over $8,000, but most models fall closer to $4,000 or so. That's where I would expect the budget to be for something like this.