Your first option is not going to work with SilverBullet, so we only have option 2.
A few things here:
You can run SilverBullet on your PC, use it locally most of the time.
In principle, you can also connect to it from your mobile phone when you’re on your local network and sync everything locally (using Sync mode, which will be the only mode in v2). The one caveat here is that for the mobile PWA version to be operational (so work without a connection to your home PC), you need to access it via HTTPS. This is a browser restriction, not of my choosing. This means that you somehow have to get your PC an SSL certificate.
The easiest way I know how to do that is to use a service like Tailscale like @marad is suggesting and is documented here: Install SilverBullet on a 64-bit Debian/Ubuntu/RaspianOS, Internet accessible via Tailscale you can ignore the LInux distribution bit there, this can be done on any machine where you can install Tailscale (also macOS).
You then have two options: you expose your local PC even outside your local network (via the funnel mode described in that topic), or just locally inside your LAN. If you expose it in funnel mode you don’t need a Tailscale client on your mobile device, otherwise you do.
What you can then do is this: open the mobile version when your PC is on to have it sync. You can then shut down your PC if you wish, or leave the home with your mobile phone. All features should remain operational without a connection (again: in sync mode). Then, when you like, you boot up your PC, open up the mobile version, and have everything sync again.