Hmm, I thought i would take the extreme, what would it take to build the smallest possible closed circuit re-breather, (closed allow for lower stored air volume than open)
An oddity of this is, the average person could survive for a day off a 60L scuba tank at the surface while idle, (11000L / 200 bar = 55L), but such a tank would be too heavy for the average person to move, so thats our baseline, so for that we have 21% as oxygen, or 2310L of oxygen, 8580L of nitrogen 3.3L of Carbon Dioxide, and 117L of argon and other gases, for at surface at rest, for an entire day, with only 660L of that oxygen actually being used,
So what do we know, for a closed circuit re-breather, you need primarily oxygen, some nitrogen, and a tiny reserve of carbon dioxide that doesn't go through the filter to keep your brain happy, and can ignore those other gases and make up the difference with nitrogen,
We breath in 0.5 Liters of, Breath out 0.5 Liters, Then The lime soda scrubber absorbs 0.03L of carbon dioxide increasing its mass, but not reducing the pressure in the loop, so your left with 16% oxygen and 84% nitrogen, (Pressure Remains the same, but the carbon dioxide being effectively scrubbed gains you an extra % of oxygen as the volume has reduced by 30mL)
so to add back in 5% of oxygen (25mL) assuming we don't have a way to isolate nitrogen, would require venting 5.24mL of Oxygen and 20mL of nitrogen, to equalize pressure, in otherwords we only get an effective use from 79% of the oxygen we tank with the simplest option possible, note this is purely focused on a rigid air volume design like what they have drawn, with a breathing bag it works a little differently, not to mention with each breath you run out of nitrogen and breathing volume as the filter fills up, assuming the bare minimum, you get 16 breaths before its kaput,
Now if we take a breathing bag style with expanding volumes, and some overhead (breath out and breath in volume with filter between, lets say 2L per bag) you have a similar breath out, the filter captures the carbon maintains the same volume, you would be at something like 19.5% oxygen on the other side of the filter, it adds in a fresh 30mL, the pressure in the loop increases by approximately 750 Pascals, (0.75Kpa) and likely expands to compensate to some degree, As such you maintain your fixed volume of nitrogen, and have many breaths before the pressure release valve on the breathing bags gets triggered, then you back-fill the lost portion of nitrogen to make up the difference on the next breath cycle, This leaves us with roughly a 150mL oxygen bottle, and from looking around generally the same size for Nitrogen or "Dilutant" bottle for 1 Hour on the surface,
Lets assume we plonk someone down for a 1 hour dive time at 10 meters, a normal recreational dive, where they would normally use a 12L tank of air at 200 Bar, So using our resting baseline that means that they use up 2400L of air in an hour, now if they spend most of there time at that 10m mark, that means the work rate is ((2400 / 2) / 11000 * 24) = 2.61 Hours per hour, or a breathing rate of 78.3mL of oxygen per breath at the surface, this more or less gives us a guideline to how big our tanks need to be,
So lets take that 150mL bottle, * 2 for the depth of 10m, then * 2.61 for the work rate, we end up with a 0.8L tank pair, or 230g of oxygen and 200g of Nitrogen, Compared to 3.1Kg of air for the 12L tank dive, sadly the tanks them self out of water still weight 1.7Kg each (3.4 + 0.43 = 3.83Kg), vs the single 13Kg (13 + 3.1 = 16.1Kg) air tank, much lighter but not something you would stick on your face until you where already underwater. This is all assuming a 200 Bar pressure.
The filter to hold the soda lime seems to come more or less around the 5Kg mark which seems to cover 4 dives, again i would assume this is for safety margin, bringing the re-breather at this low end up to 8.8Kg, then the expansion bags, should be light, and the electronics and solenoids, lets bargin 700grams for that, so 9.5Kg is about the smallest i could see such a thing being done for with a normal recreational dive in mind, too heavy to stick on a face, but quite possible to build into a backpack,
At least that is the best approach i can come up with without waving a magic wand,
There is a darker way to approach this, you can replace the nitrogen with hydrogen, as its less likely to cause nitrogen sickness at depth, which would allow electrolysis to recharge, (providing you can scrub the other nasties) but its still 3670 Watt hours to electrolyse 1L of water assuming everything goes perfectly, and to do that 1 hour dive, you need 320L electrolysed for every hour at that depth, which is way more than any battery a person could physically lift could produce, (a normal car battery only gives you about 1200 Watt Hours),
much easier to run atmosphere through a dust and air filter then carbon dioxide scrubber to recharge at the surface, you could then use a normal nitrogen membrane to seperate off the nitrogen from the oxygen, and bobs your uncle for quick and dirty (and possibly unsafe) now this part just for the air compressing would only take 120 Watt hours to pump both tanks to pressure, plus what other loads the filters and nitrogen membrane add, but more than reasonable on a sizable lithium, Set up your recharge tanks on the beach, leave it to recharge over the hour your re-breather dive takes, and do back to back dives until your filter needs replacing (probably 2-3 dives)