An incidental, token amount
The rules of silvering you present are correct, but they also mean that it costs the same to have any weapon (or ten pieces of ammunition) silvered. It will still cost 100gp, whether you need the amount of silver required for ten blowgun darts, a dagger, or a greatsword.
Since the cost of the process is the same regardless of how much material is involved, it stands to reason that the value of the "time and expertise" is considerably more than that of the silver itself. Perhaps the smith, by charging the same regardless, is making a little less profit on a greatsword than a dagger, but apparently the difference is not enough to warrant a variable rate.
Furthermore, adding this silver (a relatively soft metal, at least compared to iron or steel) to the weapon is neither reducing the weapon's effectiveness (as stated in the rules) nor increasing its weight (assumed since it does not state that the properties of the weapons change, and increased weight would certainly affect the range of ammunition).
Thus we can conclude that the actual amount of the silver is negligible compared to the "time and expertise" needed to successfully work it, and is certainly less than the initial weight of ten blowgun needles.
Comments on this answer suggest that the DM is trying to break up the standard 100gp price of silvering. It may be worth considering why.
Is it that the DM is rewarding you for role-play with the blacksmith owing you a favor? In this case the DM should decide how much of a discount on this standard service they would like you to get, and that decision should be based on how much they want to reward you, not on an accurate calculation of silver content.
Is it that the DM is willing to reduce the price for a desired service in return for a 'fetch quest' of you bringing silver to the smith? This is trope that can be usefully employed, although silver seems an odd choice as it is abundant in most game worlds. In any event, the answer to how much silver you would need becomes 'as much as I acquired in the quest set up by the DM', and again accuracy of calculation is not important.
Is it that the DM is trying to build an internally consistent economic system in the game, which can allow material values to interact with labor costs? Good on them! But I would suggest that they start with high consequence, frequently occurring interactions. If your game is not one in which you frequently are owed favors by skilled craftspeople resulting in such deals, accurate calculation of this particular exchange is unlikely to be useful in setting up the system.
If the blacksmith owes you a favor, but you have to supply the silver, any small, token amount of silver should do. Give the smith a handful of silver coins or a small silver item. This will fulfill any role-playing aspect of you providing the smith with silver without having to accurately calculate what is a negligible and ultimately meaningless price.