When refrigerated, rehydrated hCG should stay fairly potent for at least one to two months. There is conflicting information about whether hCG can survive a freeze-thaw cycle. It might be worth freezing the excess if the alternative is throwing it away. Then you have the option of trying it later to see if it is still potent.
I think realistically you're more likely to see a net loss of testosterone as a long-term result of this experiment. The first few doses of hCG may indeed give you a boost, as it supplements your existing LH. But the non-pulsatile nature of the hCG makes it more suppressive than LH, likely leading to less testosterone than at baseline as time goes on. Only at full replacement doses of hCG are you likely to see higher testosterone.
I don't know if this would work either, but a slightly better bet is to try infrequent hCG doses, e.g. once a week. This is almost five half-lives of hCG, allowing it to mostly clear out before the next dose. Maybe this would keep the suppression in check, giving you extra testosterone in the first few days after each injection. Of course it could also lead to below-baseline testosterone in the final days before the next dose, as well as lower average levels. Only experimentation would tell.
In any case, it makes sense to keep the starting doses low, 100-300 IU per injection. Suppressive effects can take weeks to fully manifest, so I wouldn't test serum levels until at least a month in.