Sure but then you have 2 full blown systems running for what is really a very simple program.
I know docker feels like another "latest thing" fad, but this kind of use case really suits it. You could run one linux instance (that can do other things) and run the 2 DSLStats in containers.
I was gonna suggest the docker thing too (someone has already posted a dockerfile for dslstats on here somewhere so they've already done all work) but you still need a base distro and I still can't think which one to recommend
Recently, I've been using elementaryOS for when I need a desktop:
https://elementary.io/ - elementary OS is a Linux distribution based on Ubuntu.
It has the apple mac style bar thing at the bottom, which, with a small bit of donkeying around, you can make the icons zoom in the same way too but I'm guessing you'll be wanting something as light as possible?
I'm guessing ESXi doesn't support docker natively? You're laughing if it does.