Generally speaking, for shorter throw, I would recommend the BenQ HT2150ST.
https://www.projectorcentral.com/BenQ-HT2150ST.htm
You can get about a 100" diagonal from 6' lens to screen. This may require the projector to be slightly forward from the back of the room, but it gives you about any size you can hope for within the confines of the space you have available.
There are not any 4K capable short throw projectors on the market for any reasonable price at this time, so it must be a 1080p projector.
I have heard that BenQ will be releasing a new 4K DLP entry level model which has the same throw distance as their older HT2050 projector that can deliver a 92" diagonal from about 8' lens to screen. This will be very welcome in the industry as it will be the shortest reasonably priced 4K projector on the market to date. It won't be available for a couple of months yet I expect. It will likely be close to $2,000, but could be around $1,500 or so. For a lot of people with BenQ's older projectors, they will jump on this model as an upgrade.