The answer to me is actually a take out double, but invoking the concept known as EQUAL LEVEL CONVERSION.

ELC is a declaration that when a takeout double gets a suit response, and doubler now removes to another suit at the same level, it is not showing extra values but simply says that the doubler is unsuitable for that suit and wishes a choice to be made between the other two.  It was a very common approach some years back but it can run into problems (say when partner gets too excited and introduces the unwanted suit at a high level, or when the opponents now bid one of your offerings) and for that reason its use has become limited.  But there are two situations where it is justified

a) it is safe when the hands are known to be limited, as here, since you can trust that your bidding will not get too high.

b) it is often needed when the takeout double is made at a higher level.   Suppose you had  ♠T3 ♥KQJ7 ♦AQJ54 ♣Q6   and you hear the bidding go  1♠-P-2♠  and you want to bid?    Or the same hand with a 2♠ opener in front of you?   Passing is way too dangerous, but if you double partner might bid clubs and leave you uncomfortable.  And partner could pass a diamond overcall with a singleton diamond and five hearts.  On both these auctions doubling and converting clubs to diamonds is the answer.

Back to today's hand : this is a candidate for a double with ELC and if you look at all four hands it seems that double hits the jackpot - but that is an illusion.  Partner will pass the double but South will not, and when South bids 2♣ things get awkward.  Partner might try a penalty double, but you have to remove that.  If you were sufficiently imaginative, you would try to maneouvre to play in spades now, but chances are you will end in 2N which is still quite acceptable.

The winners on today's hand were those who passed out the 1♠ opener, as it went down three.   It's a difficult game.

The first thing to register here is that partner's second double is still for takeout. At this level, you are not expected to take out into a failing contract, but partner is showing extra attacking (rather than defensive) values, andd is encouraging takeout.

Your values and shape are far from exciting with half your HCP in the opponent's suit, so my instinct would be to defend as taking four tricks in defence will be a lot easier than making eleven tricks yourself. But if you do bid which suit do you choose? 

The answer to that question is that you don't choose. In situations like this, unless you have a strong preference, you should be asking partner to choose, and 4NT here is just like the unusual 2N, showing two places  to play and asking partner to pick. Today 4N would get a 6♣ response from partner and now you are in the slam bid at only one table.

You are in reality not likely to get to this slam after the bidding starts as shown, which illustrates well how important it is to cramp the auction, as the 4♠ bid did. However you are likely to collect +500 against 4♠ so all is not lost.

The key is that partner must get involved, and there is only one way to do that which still allows your partnership to defend, and that is with a double.

In common with the theme that where at all feasible, any double shows extras and a desire to bid on (progressive rather than penalty) this double by a hand which has already declared weakness is an ACTION DOUBLE, giving the message that you want to bid again but you are letting partner make the final decision. Here it would allow partner to sacrifice in a very cheap 5♠. You give up the ability to set a trap for the opposition by making a deceptive bid, and then doubling for penalties, but that is a small price to pay.

It is curious to note that had you overcalled 1♠ you would have known what to do as then partner's 4♠ bid would tell you more about their hand, while the raise of 2♠ covers a wide variety of hands. Other winners on this hand were the Easts who opened 1♠ in first seat. With a good lead and this vulnerability, opening in first seat is the right thing to do. After the 1♠ opener, West had no problem bidding 5♠.