For me this depends if you switch off any device in the year, for example, in my case when I’m on holidays usually out home, I switch off the proxmox server but I don’t switch off my router so, in my case I’ll choose to install in the router.
Maybe you will ask why I switch off my proxmox and basically the answer is because if something fails and I’m not at home to solve it, I don’t want the server running wrongly until I’m back.
Which provider are you using to have the domain so cheap?