As always with performance, the only good answer is test, measure, test again, measure again ...
That being said, yes caching will probably be faster, the question is will it be significantly faster in your context.
The drawback of caching is memory usage and thread safety. MethodHandle itself seems to be immutable (not explicitly documented but looking at the contract, this is most probably the case). MethodHandles.Lookup is used to check for security constraints, so caching might allow you to bypass, or at least change the behaviour regarding security constraints. This might or might not be a concern to you.
Memory usage comes into play if you cache lots of different MethodHandle. You need to make sure your cache is bounded and sized correctly. My guess is that you will not cache a lot of different instances, but that's only a guess...