1NT as a response to 1♠ looks a little odd with all those clubs but it’s still the best thing to do. The problem with bidding 2♣ immediately is that it shows 10 points and it’s forcing partner to bid again. That means you will never get to play in 2♣.
If you really want to play in clubs there are two ways you can do so. You could bid 2♣ and then 3♣. But that would show 10 points, as discussed above. Alternatively, you could bid 1NT and then 3♣. Your 1NT bid shows 6-9 points and if you then decide to bid 3♣ later then partner will know you have a weak hand with lots of clubs.
Still, do you really want to play in clubs? Partner has shown spades and hearts and it looks like one of those hands that are going nowhere. Better to play in 2♠, simply putting partner back to his first suit and knowing you're probably going to be in a 5-2 trump fit. At least it’s only at the two level and it is in a high scoring major suit rather than a piddly minor suit.