Original lacquer - but that scratch is a bit too much (and a similar one on the opposite side too). Any easy and cheap remedy? I'm thinking of covering as much as possible leaving a tiny border untreated as it would be very tricky blending the new lacquer to the old...
So you got your answer.
Easiest solution...have it chem-bathed by a pro, then hand-polished with some sort of paste (I suggest Wenol, Maas...Flitz also is OK. If you desire a mirror-like appearance, then give it a second go with Hagerty 100....that stuff has fooled people into believing they were looking at a lacquer finish when in fact it was just nicely polished brass. It won't match, but will look better than THAT.
It WILL start to patina, so there isn't a hecka lotta control you have as to HOW it patinas. Sometimes, as it Patinas it will start moving towards the color of the orig lacq so not look bare anymore. When it gets close, you can use a wax to sorta quasi-arrest the patina (something as simple as Pledge furniture polish, or less simple like a metal wax).
Next, you could investigate buying metal tinters, then apply the tint then wax it. Jax Gold is one, it has a so-so success rate tho. I use tints marketed for outdoor metal sculptures, light heat torch surface then brush applied.
These will not, again, MATCH the hue of the original lacq, but will get it sorta close enough in tint so that from 10 to 12 feet away, there isn't much to notice.
Last possibility - SPOIT LACQUER (By a tech/pro, please). There are lacquer products available to a tech where they mix up lacq with tint, get an OK color-match to original lacq, then usually brush apply. It gets pretty darn close initially and will LOOK really pretty decent...until the lacq starts to age over time and the two lacq colors start falling out of similar hue as time goes by.
NOT many techs do this anymore, honestly. But results are good for a while (could be anywhere between 6 months -5 years).
****DO NOT DO A D.I.Y. SPRAY-CAN SOLUTION ! NEVER ! It will look bad in a pretty short time, as in less than a year tops, and on a horn like a 10M, 90% of sax folk will agree it would have been better off just left bare.