I have been on this quest for several years... still havent found the perfect scope, but so far my 'keepers' are;
Lrhs 3-12
Lrts 4.5-18
Klassik 3-12x50 ASV
Nxs 2.5-10x32
My advice? Try to find a used Bushnell LRHS 3-12x44. Only gripe; they seem slightly overweight, but that *might* just be the cost of reliability... will come in under budget (if you can find one).
Klassik will meet your weight requirement, but even used comes in slightly over budget. Gripes; these have limited elevation travel and fixed parallax.
I find it ironic that the Klassic is the only model on my list still in production, and yet it is oldest design of all
Good luck in your quest!