As far as my understanding goes, bio processes using whole organisms/cells like algae/bacteriae/fungi (as opposed to processes using a small set of enzymes or plain chemical processes using catalysts) inevitably come with the disadvantage of heaving a huge overhead. The organism/cell is a complex machinery where there's hundreds of pathways and chemical reactions running that are necessary to just keep the organism alive. Only a fraction of the energy input comes out in the form of the desired molecule. Most is dissipated as heat.
Maybe not the worst analogy would be the comparison between two systems for mining cryptocurrency: A, a specialized chip (ASIC), specifically designed to efficiently run the required algorithms and B, a general purpose PC that has all kinds of peripherals (monitor, keyboard, mouse, hard drive, ...) attached to it, drawing additional power that doesn't benefit the mining.
B of course will never beat A in terms of energy efficiency.
So I'd guess if there's a purely chemical/enzymatic process that can do the job, it'll be that, rather than a whole cell/organism.
> In theory would only need sunlight and water to operate ...
Don't forget minerals.