I think opening silver/x is usually better (adjust for the board obviously), and then hope to get a good overpay on t3 or t4 (or later)
Hm. That is also an option I haven't thought of. But it is in general wise to overpay on 3/4 when you open Silver/Doctor?
You will leave 5 out of 6 cards on deck: 3 Coppers, 2 Estates and the $3 you bought. If the $3 is Ambassador, that may be a good idea, because you have almost ensured returning two Estates (only bad case is Amb on top and Estate on the bottom). You are always ensuring no collision with Amb and Doctor and given your opening, even returning one Estate and buying Silver is not such a bad deal.
However, with other $3s, that may be a bad deal. Especially with Silver, the chances to get to $5 in T3 are not great, and not so great for T4 either. Moreover, T4 gets you the $5 half a shuffle later due to the early shuffling Doctor provoked. I think if your $3 is a power card you want to be playing more (I can only think of Ambassador here, possibly Swindler?), triggering a shuffle on T2 may be a good idea, but it does not look smart otherwise.
If there is some cheap draw (Smithy, Oracle, Envoy), you are probably best getting money and trying to get a good overpay during the second shuffle to compensate the delayed Doctor buy.