The second option requires a bit more time and effort than a 301 redirect. You could do a mini overhaul of the site and turn it into a microsite for your main domain. This option is good for exact-match domains for your targeted keyword, and there are other reasons for going the microsite route that Rand's highlighted in his post about root domains, subdomains, subfolders and microsites. This strategy also works better if the old domain has decent rankings for the keywords you're targeting.
Also, I currently have two other domains that are registered with Squarespace, and I have their annual payments set up automatically so I never have to manually pay (and so nobody else gets that window of opportunity to take the domain away from me). Would it be possible that the owner of this domain might just have a similar system in place and that I might be wasting my time?
The advanced search feature lets you limit by domain extension What URL Domain Extensions Stand For and Why They Are Needed What URL Domain Extensions Stand For and Why They Are Needed There's a lot more to the internet that just .com, .org, and .net sites. The world of top-level domains exploded a few years ago. But what is a TLD? Let's find out. Read More , age, number of characters, and type. Types include expiring, auction, buy now, and more.

See They start it in auction listings after it expires and before pending delete. I don’t know the exact dates, but if you monitor it you’ll see it go from auction, to closeout, to drop. You’ll need an account (I believe $5 per year) to bid and buy.
If you buy a dropped domain from Snap or Namejet, the backlinks seem to be worthless for SEO.  They are valuable for traffic if it's targeted to your site.  Go ahead and 301 redirect into your site because it's the traffic from the back links that is worth something.  I use the Google URL builder to redirect these names so you can see the domain the traffic is coming from.
The whitelist filter for wanted characters is now split into three filters. The old whitelist is now named whitelist (only). When used, the domain has to consist only of the defined characters or numbers. No other characters are allowed. The new whitelist (any) filter finds domains that have at least one of the defined characters or numbers. The new whitelist (all) filter finds domains that have all defined characters or numbers at least once.