If you don’t want to mess with another VPS you can use a global server load balancer (GSLB) provider like Akamai, Cloudflare, Azure, etc.
This being a self-host community though it’s unlikely you’d want to pursue something like this, but without knowing more about your specific use case it’s tough to make a recommendation.
If global high-availability is your primary goal then a hosted solution is probably best.
If this is just an exercise you and your friend are working on for giggles and it’s not for a mission-critical Production instance, then presumably self-hosting a load-balancer on each of your servers that includes both nodes in a target group would achieve this, though somewhat counterintuitive; if the website goes down at either location, I would imagine there’s a pretty high likelihood the LB itself would be down as well.
Where would you host the script? If it’s expected that the server that fires it off is always online and performing health checks, why not have it host a load-balancer? Or another local instance of the website? It’s something fun to play around with, but if this is for anything beyond a fun exercise there are much better ways to accomplish this.