TL;DR: You can't make a large lens to increase the brightness, but you can increase your depth of field by focusing at or beyond the hyperfocal distance.
You didn't say what you were photographing, but with anthotype, cyanotype, etc., I imagine you're probably just trying to get basic photos of mostly static things. Landscapes/cityscapes, pictures of buildings, etc., because of the very low ISO and hence long exposure time requirements. Most likely not macro subjects.
Intended subject, or more specifically, subject distance from the camera, is important if depth of field is a concern. If your subjects are what I've assumed they are, make yourself familiar with the concept of hyerpfocal distance, the camera-to-subject distance where depth of field is mathematically infinite. When your subject is at the hyperfocal distance, everything between half the hyperfocal distance to "infinity" is within your depth of field. The hyperfocal distance H is given by,
H = ƒ2 / Nc
where ƒ is the lens's focal length; N is the ƒ-number (relative aperture) of the lens; and c is the circle of confusion diameter, sort of a quantifier of desired resolution or acuity. For 35mm film cameras, c = 0.03 mm is a good estimate. For higher megapixel crop- and smaller sensor digital cameras, values as low as c = 0.005 mm are stated.
Aside: Just because hyperfocal distance is a thing, doesn't mean that you should always shoot for it (at least, given a regular camera with ISO options). Personally, shooting with a DSLR or mirrorless camera, I consider hyperfocal distance as a mathematical option, but nothing that I aim for. I focus on my subject, and tradeoff shutter speed and aperture as needed to achieve the look that I'm seeking. But if your goal is to maximize depth of field, hyperfocal distance focusing is a starting point.
As far as image brightness is concerned, understand that photometric exposure, sort of the amount of photons available to capture on your film or image sensor (given constant illumination from the scene), is determined by only two factors: relative aperture N (ƒ number), and exposure time t (shutter speed). Specifically, exposure value, EV, is defined by:
EV = log2(N2 / t)
See EV at Wikipedia
This means that for a given combination of t and N, if you change, say, t by a factor of 2 (i.e., halving exposure time from 60 seconds to 30 seconds), to have the same exposure on your film, you need to compensate by decreasing N by a factor of 1.414... (i.e., square root of 2). This is the fundamental tradeoff between relative aperture and exposure time.
Regarding your question,
Is it optically possible for a custom-built camera to form an image with both a reasonably deep depth of field and a very bright image (lots of light gathering)? Is it simply a matter of lens shape and size and camera body dimensions?
The fastest lens that is mathematically (but probably not physically) possible is ƒ/0.5.Note. This is due to the conservation of étendue. It's a difficult concept to understand, but see the following questions and links:
See also étendue at Wikipedia. Also, the XKCD What If? explainer, Fire From Moonlight: Can you use a magnifying glass and moonlight to light a fire? is very salient to your question. When you said,
Thus, no matter what sort of lens you construct, you will never be able to have a relative aperture faster than ƒ/0.5. The fastest production lens was the Carl Zeiss ƒ/0.7. Only 10 were ever made – six were sold to NASA, Zeiss kept one for himself, and three were bought by Stanley Kubrick for use in filming Barry Lyndon (see: 1, 2). There are some ƒ/0.95 lenses, such as the Canon 50mm ƒ/0.95, Leica Noctilux 50mm f/0.95 ASPH. Beyond that, expect no better than ƒ/1.2 or so.
While the superfast lenses of ƒ/0.95 – ƒ/1.2 are renowned for their beautiful shallow depth of field, you can still use them to photograph subjects at the hyperfocal distance as noted earlier.
Note: It is possible to have a lens faster than ƒ/0.5, but it won't have a combined light transmission (i.e. T-stop) faster than T/0.5. That is, the real light loss of the lens, given by its transmission ratio (less than 1, around 0.7–0.8 for uncoated lenses; upwards of 0.96 for lenses with antifreflective coatings), can be modeled as an ideal lossless lens with a fixed neutral-density filter built-in. The combination of geometric aperture (represented by ƒ-number) and built-in "ND filter loss" characteristic (T-stop) will never be less than 0.5. The reason for this is because ƒ-number N is really a consequence of the lens's numerical aperture,
N = 1 / 1/(2×sin(𝛼/2))
where 𝛼 is the acceptance angle of the optical system. The maximum value of the sine function is 1, which happens when its argument is 90º. This doesn't say the every, or perhaps even any optical system can have an acceptance angle of 180º, this just puts a hard upper limit on the numerical aperture, and thus, a maximum "speed" on any optical system.
When I said the maximum T-number of the lens system was 0.5, implicit in that is the assumption that we're talking about an image-forming optical system. That is, we're interested in photographs, sharp images with low or minimal aberrations, distortions, etc. It is possible to have ƒ-numbers somewhat lower than 0.5, but the system is no longer usefully image-forming. This comes back to the étendue-conserving "you can't squoosh light onto a smaller area" XCKD What If? explainer.