I've seen/owned/handled a lot of Martini rifles in my time and that's the first one I've seen that isn't threaded into the receiver.
From your photos, the barrel appears to be screwed to the block, attached to the receiver.
Nice looking rifle.
I recently put a new take off barrel onto a small frame Martini. I fitted a sleeve over the barrel by sweating it on. The sleeve has two grooves milled on the top to accept 10mm scope clamp type bases.
It had a similar ejection problem that wasn't related to a burr. The chamber was pitted and the bore wasn't great either
Many of those lovely little rifles were ridden hard and put away wet.
90 years ago, there may have even been some black powder cartridges used, especially during the depression and during WWII, when any 22rf ammo was almost impossible to find.
From your description, your bore is nice and likely so is the chamber. If there is a burr on the bottom, it's very easily removed with a fine tooth, small round file.
If you want to be really picky, you could take it to a smith and he will either file it off or run a reamer into the chamber to clean it up.