It's only a support double if you have agreed this with partner; no mention is made of this agreement so presumably it doesn't exist. So double just shows cards, hopefully 13 of them and hopefully some with pictures on them. The problem now is that you have no idea of the kind of hand that partner can have. 4S could be playable or just plain silly, and 4D could be close to making or down a couple. Which is one argument for bidding 4S directly - you have made your decision (aka "guess") and the opponents now have to decide what to do. Doubling 5D, if it is bid, looks reasonable. However, that is not now the problem. It would help to know partner's overcalling style here. 4S is entirely reasonable at this point, but if partner is known to make, er, enterprising overcalls, it may well be right to pass. Give partner KJxxx Kxxx xx xx, and on BBO we have all seen players bid 1S on less, 4D could be close.