Congrats on the new watch, sorry to hear it has been scratched.
I have a TopGun Double Chrono which had a few scratches on it and it was recently sent back "home" to have an unrelated issue fixed with it. Whilst it was there, I was offered a case refinishing. My initial thoughts were "but its ceramic, what the hell are they going to do to it to get the scratches out - it is almost impossible to get the scratches in in the first place so it must be even harder to get them out!" but for what they were quoting for the job (a very reasonable amount) I thought what the heck I will let them try and see what comes back.
When I got the watch back, it was like new, no sign of any of the scratches / marks that were on the case and they even got rid of a scratch on the iron ring around the crystal.
My point here is, based on the fact that you have already scratched it you are more than likely to get a few more over the course of the life of the watch. If you cant get the scratch out yourself with something like like an eraser then dont go to too much effort as you may do more damage than good. Then, when the watch is 2, send it back for its scheduled grease and oil change and at the same time, get the case refinished - by the experts. I have just sent another watch in for a full service and case refinishing is part of the "standard" full service (as well as a crystal replacement)- I am not sure if this only applies to vintage watches or all watches that get a full service.
Cheers,
Ben