Engineering RuBisCO (the enzyme in charge of carbon fixation and the rate-limiting step in photosynthesis) has been quite difficult, as some side reactions generate products that, for instance, inhibit RuBisCO itself. If you make the fixation reaction more efficient the other ones follow.

Another option is increase the amount of RuBisCO in the cell. In cyanobacteria the enzyme is encapsulated in compartments called carboxysomes. Some studies have introduced carboxysome-like components in non-cyanobacterial organisms with varying results.

They live in a hugely competitive environment. Any custom algae that weren't as competitive as natural ones would soon die out, and it's pretty hard to beat nature at its own game...