I am not totally familiar with the AE-1, but most of the SLRs are capable of shooting in manual only and manual focus with the correct setting. At one time I had a Nikon FM that Steve mentioned and it is indeed fully manual. It takes one or two small battery cells for the light meter, but otherwise no batteries. The Pentax K1000 is similar. If you are bottom fishing there are a bunch of older cameras that might fit you bill.
I'm speaking from experience from a few years ago, but when you get into manual focus (not auto focus) and "prime" (not zoom) lenses there are some high quality lenses in the bottom fishing price range. If you go back a little further into SLR development you might get into screw mount lenses (as opposed to modern bayonet mounts) that are probably even cheaper. Then, there was the era where rangefinders with fixed lenses were common and lenses could not be interchanged. There's a little artistic niche I've been into where not only are you shooting totally manual, but is only one choice for the lens focal length. The further back you get, you get into cameras with fixed lenses and also fairly crude methods of focus and exposure. I have had used cameras from the 30's and 40's where you have to gestimate the distance to the subject or use a separate range finder, as well as handheld light meters. It all depends on how far back into analog days you want to go. ...and then there are pinhole cameras that don't even have a lens or shutter