I have several mosins. Only my 91/30 is scoped. M44 recoil will knock cheap scopes out of whack every time. Not reliable, unless you spend the big bucks for a high end set-up. Even then, I am not sure there are LER choices out there that are designed for such recoil. As stated the 44 will shoot better with iron sights and the bayo extended, as it was designed to do. It was not intended to be an accurate long range shooter.
That said, if you want to try scoping it, I have had excellent results on my long rifle with a cheepo nc star long eye relief 4-12 zoom. Optics Planet had them for $25 or so. If you heat up the standard leaf sight base to melt the silver solder underneath, and gently tap the base with a brass hammer, you will find a nice dovetail mount under there. Appox 3/8", if I recall. Easy enough to find a scope mount that size. I got one from Wally World for about $5. Total expense was about $40. Eye relief is perfect, and it has a bit of parallax, but otherwise a surprisingly workable scope. After several hundred rounds, the mosin remains on 200m zero, and it shoots better than my Dad's Swede sniper. Consistent 1-2" at 200m. Different rifle with a very tame kick. If the recoil of the M44 is too much for the scope, it's easy to put it back to stock and you haven't spent much. Have fun!