Website reloading before Safari Extension saves suggested password

It's possible to separate the keys (and other items) into different vaults (work, personal, whatever). And then configure the 1Password SSH Agent via the config file to allow access to keys in specific vaults:
https://developer.1password.com/docs/ssh/agent/config

This would allow you to configure your work machine to only look at the keys in your work vault.

I only put work stuff in my work vault, which is accessed from my work laptop. Personal stuff goes into my personal vault, which is accessed from my personal machine.

Well, I kind of mean that - but configurable from 1Password's side.

I have several SSH keys in 1Password, but on my work laptop, I only want my "work" SSH key available. I'd prefer to keep the key within 1Password. But I don't think it's possible to tell 1Password to only use a specific key on a specific host (as in, it should ignore all other keys when using SSH from a specific computer).

Not sure this if this what you're asking about, but I use many SSH keys with 1Password. It's possible to control which SSH key is used for which account (user@host) by configuring them in ~/.ssh/config file. In fact, this isn't a feature specific to 1Password. It's how SSH works.

For more details on the ~/.ssh/config file:
https://linux.die.net/man/5/ssh_config

I typically specify keys using IdentityFile option in ~/.ssh/config to identify the SSH public key I'm using for a specific user and host, with the private key stored securely in 1Password.

Depending on what you're looking to do, it might be possible using ~/.ssh/config.