Honestly - I would recommend you get a spot instance on AWS and learn to manage your own server. Full control over everything, which mods, when to back up, etc. But also full control over resource contention. While AWS does share its infrastructure between clients, youre not going to have the degree of either CPU or network contention you get while budget gaming hosts.
As long as you remember to shut down your server between gaming sessions, it shouldn't be that much more expensive that a game host.
Of course, if you have no seever admin skills and no desire to learn, it might not be for you.