I had the exact same problem back in 2012 (or was it 2013?). The worst is when you have a fish on and the reel starts to free spool! I've taken the anti-reverse bearing out and cleaned it. I can't remember if I just used WD40 or isopropyl alcohol? That seems to have done the trick - mind you I don't use my Luna 253 very much anymore. I use it on the odd occasion for springs and it's held up well since.
So, you can try and clean it out. And as you mentioned, check out Mike's Reel Repair to see if they have the replacement part. You can try and do a search for the part number (F55-2703) online and you should find this:
https://www.mikesreelrepair.com/f55-2703-roller-clutch-nla/The Luna is a great reel, but my issues with the anti-reverse bearing has soured my experience with it a bit. I tend to use my Calcutta 200B as my go-to, or my centrepin set-up. Good luck!
EDIT: Whoops! Looks like the part is no longer available? Maybe see if they can recommend an alternative or upgrade?