There are all sorts of remedies that will work well.
Not just Martins but just about every other 22 rimfire out there can be plagued with a similar issue. The OP's rifle is easily fixed with all of the above mentioned methods.
I replied to him with a PM.
I have removed dozens of similar burrs from as many rimfire firearms.
I use two methods. If the rifle has a a take down feature such as the OP's then it makes the job easier.
There can be issues with "ironing" out the defect. The iron can expand the chamber mouth and just pushes the burr back into the hole. It can also spread out the divot in a manner that will leave a sharp edge.
Also, the divot is loose in the hole and can cause issue later.
I just find it easier and less worrisome to remove it completely.
YES, I most often use a FILE. A couple of round chainsaw files work well. Start with a 5/32 diameter and very carefully remove the burr without enlarging the chamber mouth. BE CAREFUL.
This isn't rocket science and if you only have a one off to do a bit of prep time and CARE will do a fine job.
I made an ironing tool and even though it works I find it requires much more care than a file. As for using a reamer, the reamer may have issues entering the chamber depending on its form. My 22 chamber reamer has a rather sharp edge leade and I wouldn't chance it on an unreduced burr. It's home made but has cut a lot of chambers.
I have measured many different 22rf case diameters. Most are between .221 and .224. A Russian TOZ3 I have has a .228 chamber diameter, most are slightly tighter.
My reamer cuts a .225 chamber.
Back to the topic. Remove the burr if you feel you are handy enough to do it. If not, take it to or send it to a smith. DO NOT use a rotary grinder. They are way to difficult to control and can get away from you easily, ruining the chamber completely.
I can also see from your photos that there are marks on the fired case being extracted from the chamber. I am surprised there aren't any issues with inserting the cartridge.