Here are two possible ideas for a solution.
The first is that rather than target a "weight range", have some tunable parameters that are correlated with the likely weight of the result. Then insert values for those parameters, and pick your combination of items, pluses, etc.
The second is to go through an iterative generative process. Make it look like this. On each iteration you randomly pick a trait to modify out of (prefix, prefix2, plus, weapon_type, suffix). If you pick a prefix2 and you have one, you randomly decide whether to regenerate the existing or generate a new one. (If you have 2 already you repeat this decision making process with the third.) Then you make a random choice of that component. (For a prefix2 make a valid choice be "none".) If adding (replacing if need be) that characteristic makes your weapon closer to the target weight, do so.
Continue this process until the target weight is reached, or until a fixed number of rounds has passed.
In essence this is a genetic algorithm, where you're trying to evolve a random weapon that fits your target weight.
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