A good explanation of the issues with 125 gr Magnum ammo in S&W K-Frame revolvers can be found here:
https://www.gunblast.com/Butch_MagnumLoads.htm
As to the "flat spots" on the forcing cones, every K-Frame revolver I've ever owned or handled which had a one piece barrel had a "flat spot" ground on the outside of the forcing cone at the six o' clock position as this was necessary to allow the yoke sufficient clearance to close the cylinder. This includes the M13-4 and 10-5 that I currently own, a 66-2 that I previously owned, and a M15 and M65 that my father owns (don't remember the dash numbers). The only K-Frame revolver I've ever personally handled which did not have the "flat spot" on the forcing cone is my sister's M64 which has the newer two-piece barrel that, as I understand it, is common to the newly produced M66 and M19.
While I did shoot a very limited amount of the very ammunition the OP has acquired (less than 100 rounds) through my M66-2 with no apparent ill-effect, my own personal policy has been to limit my K-Frame Magnums to either .38 Special ammunition or .357 Magnum ammunition with bullets no lighter than 140 gr. My M13 and 66 as well as my father's M65 have been shot nearly exclusively with 140+ gr Magnums (mainly 158's) for years and have no problems at all.
Honestly, full-power .357 Magnum ammo with light bullets is going to be harder on any revolver than other ammunition would be. You have a relatively light-for-caliber bullet with a big charge of slow-burning powder behind it and that's a recipe for forcing cone erosion and flame cutting. I've "dissected" a few rounds of both Remington and Federal .357 Magnum ammunition and they were all loaded with powder that looked nearly identical to H110/W296 and the 125 gr Remington JSP had, IIRC, around 22 grains of it.
The reason, I think, that we don't see as much issues with 125 gr Magnums from other "medium frame" revolvers is because most are bigger and have thicker forcing cones with no "flat spot" like the K-Frames do. Guns like the S&W L-Frames, Ruger GP100, and Colt E, I, and V frames are all more along the lines of what I would consider a ".41 Caliber" frame size while smaller guns like the S&W K-Frames and Colt D-Frames are more what I'd consider a true ".38 Caliber" frame size. These larger ".41 Frames" allow for a taller frame window and thus a thicker forcing cone when chambered for .357 Magnum or other .35 caliber cartridges like .38 S&W, .38 Special, or 9mm. Of course, true large frame guns like S&W N-Frames, Ruger Redhawks, and Colt New Services were designed for even larger .44/.45 caliber cartridges and thus have room for very thick forcing cones when chambered for .35's.
I suspect that even the larger frame guns would eventually show flame cutting and forcing cone erosion faster with full-power 125 gr Magnums faster than they would with heavier bullets, but they're over-built enough that their owners either never shoot enough to cause problems or they develop other issues like timing problems or endshake first. On the other end of the spectrum, small-frame guns like S&W J-Frames, Ruger SP101s, Ruger LCRs, and Colt Magnum Carrys probably aren't typically shot with enough magnum ammo to have problems, regardless of bullet weight, due to their stout recoil. Basically I think that the problems are most well-known in K-Frames because they're the smallest, lightest guns that most people can stand to shoot enough Magnum ammunition from to have issues.
Because the OP stated that he had enough other .357's to shoot his 125 gr Remingtons in, I'd recommend that he do that and reserve his M65 to .38 Specials or .357 Magnums with bullets of 140 gr or heavier. There are a lot of very good .357 Magnum loadings with heavier bullets and there really isn't anything you can do with a 125 gr .357 Magnum that you can't do with a heavier bullet in that caliber. He might be able to get away with shooting his Remington JSP's in his M65 without problems, but given that older K-Frame Magnums like his are increasingly valuable and that replacement barrels for them are no longer made, why risk it?
https://www.gunblast.com/Butch_MagnumLoads.htm
As to the "flat spots" on the forcing cones, every K-Frame revolver I've ever owned or handled which had a one piece barrel had a "flat spot" ground on the outside of the forcing cone at the six o' clock position as this was necessary to allow the yoke sufficient clearance to close the cylinder. This includes the M13-4 and 10-5 that I currently own, a 66-2 that I previously owned, and a M15 and M65 that my father owns (don't remember the dash numbers). The only K-Frame revolver I've ever personally handled which did not have the "flat spot" on the forcing cone is my sister's M64 which has the newer two-piece barrel that, as I understand it, is common to the newly produced M66 and M19.
While I did shoot a very limited amount of the very ammunition the OP has acquired (less than 100 rounds) through my M66-2 with no apparent ill-effect, my own personal policy has been to limit my K-Frame Magnums to either .38 Special ammunition or .357 Magnum ammunition with bullets no lighter than 140 gr. My M13 and 66 as well as my father's M65 have been shot nearly exclusively with 140+ gr Magnums (mainly 158's) for years and have no problems at all.
Honestly, full-power .357 Magnum ammo with light bullets is going to be harder on any revolver than other ammunition would be. You have a relatively light-for-caliber bullet with a big charge of slow-burning powder behind it and that's a recipe for forcing cone erosion and flame cutting. I've "dissected" a few rounds of both Remington and Federal .357 Magnum ammunition and they were all loaded with powder that looked nearly identical to H110/W296 and the 125 gr Remington JSP had, IIRC, around 22 grains of it.
The reason, I think, that we don't see as much issues with 125 gr Magnums from other "medium frame" revolvers is because most are bigger and have thicker forcing cones with no "flat spot" like the K-Frames do. Guns like the S&W L-Frames, Ruger GP100, and Colt E, I, and V frames are all more along the lines of what I would consider a ".41 Caliber" frame size while smaller guns like the S&W K-Frames and Colt D-Frames are more what I'd consider a true ".38 Caliber" frame size. These larger ".41 Frames" allow for a taller frame window and thus a thicker forcing cone when chambered for .357 Magnum or other .35 caliber cartridges like .38 S&W, .38 Special, or 9mm. Of course, true large frame guns like S&W N-Frames, Ruger Redhawks, and Colt New Services were designed for even larger .44/.45 caliber cartridges and thus have room for very thick forcing cones when chambered for .35's.
I suspect that even the larger frame guns would eventually show flame cutting and forcing cone erosion faster with full-power 125 gr Magnums faster than they would with heavier bullets, but they're over-built enough that their owners either never shoot enough to cause problems or they develop other issues like timing problems or endshake first. On the other end of the spectrum, small-frame guns like S&W J-Frames, Ruger SP101s, Ruger LCRs, and Colt Magnum Carrys probably aren't typically shot with enough magnum ammo to have problems, regardless of bullet weight, due to their stout recoil. Basically I think that the problems are most well-known in K-Frames because they're the smallest, lightest guns that most people can stand to shoot enough Magnum ammunition from to have issues.
Because the OP stated that he had enough other .357's to shoot his 125 gr Remingtons in, I'd recommend that he do that and reserve his M65 to .38 Specials or .357 Magnums with bullets of 140 gr or heavier. There are a lot of very good .357 Magnum loadings with heavier bullets and there really isn't anything you can do with a 125 gr .357 Magnum that you can't do with a heavier bullet in that caliber. He might be able to get away with shooting his Remington JSP's in his M65 without problems, but given that older K-Frame Magnums like his are increasingly valuable and that replacement barrels for them are no longer made, why risk it?