Using IndexedDB, the forthcoming SQLite support in Chrome, or the browser filesystem components all might be good paths. Either way, what do you think about using WASI instead of Linux? I think it would reduce the overhead significantly.
I'm not familiar with WASI but I'll take a look. We're using Buildroot now and it's nice and small, but we could make it smaller by optimizing the kernel. Now, if I only know how to optimize the kernel without breaking things :)
Where's all those kernel hackers? Your help, we need. :)