1080P will be fine for a documentary, though if you were doing a "Planet Earth" type landscape documentary, 4K might be better. But for a typical talking heads + interior activity doc, 1080P will be fine unless you are specifically trying to make it for Netflix, Amazon, and Hulu, who need 4K content so may take an extra interest just because you have a 4K version.
You may want to test the best method of downscaling 4K to 1080P -- you might find that you can do it better on your computer than what the camera does, particularly if you can record uncompressed 4K to an external recorder, so for particular shots where having a 4K image to play with in post, you could shoot those portions in 4K and then downscale to 1080P to cut into the rest of the doc.
You may also want to find the best way of recording to get the highest data rate as possible. Unfortunately the camera only records and outputs 8-bit 4:2:2 or 4:2:0 but at least with external recorders, you can avoid compression. Again, it depends on the nature of the documentary and how much color-correction you think it will need. If the budget is really tiny, you may have to live with whatever the highest quality (lowest compression) that the camera can record internally.
Also, keep in mind that for a theatrical DCP, you may have to add tiny, tiny side borders to create a 16x9 within 1.85 2K DCP (1920 x 1080 inside 1998 x 1080), so you might need to create that version as well as a 1080P version (perhaps on blu-ray for some small festivals.)
Content is king with documentaries, so if you can deliver quality 1080P footage, I think you'll be fine with whatever screening venues you are aiming for.